\n",
+ " \n",
+ " ![\"QuantEcon\"](\"https://assets.quantecon.org/img/qe-menubar-logo.svg\")
\n",
+ " \n",
+ "
\n",
+ "\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3caf97ca",
+ "metadata": {},
+ "source": [
+ "# Geometric Series for Elementary Economics"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f4317903",
+ "metadata": {},
+ "source": [
+ "## Overview\n",
+ "\n",
+ "The lecture describes important ideas in economics that use the mathematics of geometric series.\n",
+ "\n",
+ "Among these are\n",
+ "\n",
+ "- the Keynesian **multiplier** \n",
+ "- the money **multiplier** that prevails in fractional reserve banking\n",
+ " systems \n",
+ "- interest rates and present values of streams of payouts from assets \n",
+ "\n",
+ "\n",
+ "(As we shall see below, the term **multiplier** comes down to meaning **sum of a convergent geometric series**)\n",
+ "\n",
+ "These and other applications prove the truth of the wise crack that\n",
+ "\n",
+ "> “In economics, a little knowledge of geometric series goes a long way.”\n",
+ "\n",
+ "\n",
+ "Below we’ll use the following imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3051c904",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "plt.rcParams[\"figure.figsize\"] = (11, 5) #set default figure size\n",
+ "import numpy as np\n",
+ "import sympy as sym\n",
+ "from sympy import init_printing\n",
+ "from matplotlib import cm"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bd0ec460",
+ "metadata": {},
+ "source": [
+ "## Key formulas\n",
+ "\n",
+ "To start, let $ c $ be a real number that lies strictly between\n",
+ "$ -1 $ and $ 1 $.\n",
+ "\n",
+ "- We often write this as $ c \\in (-1,1) $. \n",
+ "- Here $ (-1,1) $ denotes the collection of all real numbers that\n",
+ " are strictly less than $ 1 $ and strictly greater than $ -1 $. \n",
+ "- The symbol $ \\in $ means *in* or *belongs to the set after the symbol*. \n",
+ "\n",
+ "\n",
+ "We want to evaluate geometric series of two types – infinite and finite."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dca1d8df",
+ "metadata": {},
+ "source": [
+ "### Infinite geometric series\n",
+ "\n",
+ "The first type of geometric that interests us is the infinite series\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots\n",
+ "$$\n",
+ "\n",
+ "Where $ \\cdots $ means that the series continues without end.\n",
+ "\n",
+ "The key formula is\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots = \\frac{1}{1 -c } \\tag{10.1}\n",
+ "$$\n",
+ "\n",
+ "To prove key formula [(10.1)](#equation-infinite), multiply both sides by $ (1-c) $ and verify\n",
+ "that if $ c \\in (-1,1) $, then the outcome is the\n",
+ "equation $ 1 = 1 $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3e6d914d",
+ "metadata": {},
+ "source": [
+ "### Finite geometric series\n",
+ "\n",
+ "The second series that interests us is the finite geometric series\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots + c^T\n",
+ "$$\n",
+ "\n",
+ "where $ T $ is a positive integer.\n",
+ "\n",
+ "The key formula here is\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots + c^T = \\frac{1 - c^{T+1}}{1-c}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bcaf6975",
+ "metadata": {},
+ "source": [
+ "### \n",
+ "\n",
+ "The above formula works for any value of the scalar\n",
+ "$ c $. We don’t have to restrict $ c $ to be in the\n",
+ "set $ (-1,1) $.\n",
+ "\n",
+ "We now move on to describe some famous economic applications of\n",
+ "geometric series."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "171870f3",
+ "metadata": {},
+ "source": [
+ "## Example: The Money Multiplier in Fractional Reserve Banking\n",
+ "\n",
+ "In a fractional reserve banking system, banks hold only a fraction\n",
+ "$ r \\in (0,1) $ of cash behind each **deposit receipt** that they\n",
+ "issue\n",
+ "\n",
+ "- In recent times \n",
+ " - cash consists of pieces of paper issued by the government and\n",
+ " called dollars or pounds or $ \\ldots $ \n",
+ " - a *deposit* is a balance in a checking or savings account that\n",
+ " entitles the owner to ask the bank for immediate payment in cash \n",
+ "- When the UK and France and the US were on either a gold or silver\n",
+ " standard (before 1914, for example) \n",
+ " - cash was a gold or silver coin \n",
+ " - a *deposit receipt* was a *bank note* that the bank promised to\n",
+ " convert into gold or silver on demand; (sometimes it was also a\n",
+ " checking or savings account balance) \n",
+ "\n",
+ "\n",
+ "Economists and financiers often define the **supply of money** as an\n",
+ "economy-wide sum of **cash** plus **deposits**.\n",
+ "\n",
+ "In a **fractional reserve banking system** (one in which the reserve\n",
+ "ratio $ r $ satisfies $ 0 < r < 1 $), **banks create money** by issuing deposits *backed* by fractional reserves plus loans that they make to their customers.\n",
+ "\n",
+ "A geometric series is a key tool for understanding how banks create\n",
+ "money (i.e., deposits) in a fractional reserve system.\n",
+ "\n",
+ "The geometric series formula [(10.1)](#equation-infinite) is at the heart of the classic model of the money creation process – one that leads us to the celebrated\n",
+ "**money multiplier**."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "94d7ef0a",
+ "metadata": {},
+ "source": [
+ "### A simple model\n",
+ "\n",
+ "There is a set of banks named $ i = 0, 1, 2, \\ldots $.\n",
+ "\n",
+ "Bank $ i $’s loans $ L_i $, deposits $ D_i $, and\n",
+ "reserves $ R_i $ must satisfy the balance sheet equation (because\n",
+ "**balance sheets balance**):\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "L_i + R_i = D_i \\tag{10.2}\n",
+ "$$\n",
+ "\n",
+ "The left side of the above equation is the sum of the bank’s **assets**,\n",
+ "namely, the loans $ L_i $ it has outstanding plus its reserves of\n",
+ "cash $ R_i $.\n",
+ "\n",
+ "The right side records bank $ i $’s liabilities,\n",
+ "namely, the deposits $ D_i $ held by its depositors; these are\n",
+ "IOU’s from the bank to its depositors in the form of either checking\n",
+ "accounts or savings accounts (or before 1914, bank notes issued by a\n",
+ "bank stating promises to redeem notes for gold or silver on demand).\n",
+ "\n",
+ "Each bank $ i $ sets its reserves to satisfy the equation\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "R_i = r D_i \\tag{10.3}\n",
+ "$$\n",
+ "\n",
+ "where $ r \\in (0,1) $ is its **reserve-deposit ratio** or **reserve\n",
+ "ratio** for short\n",
+ "\n",
+ "- the reserve ratio is either set by a government or chosen by banks\n",
+ " for precautionary reasons \n",
+ "\n",
+ "\n",
+ "Next we add a theory stating that bank $ i+1 $’s deposits depend\n",
+ "entirely on loans made by bank $ i $, namely\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "D_{i+1} = L_i \\tag{10.4}\n",
+ "$$\n",
+ "\n",
+ "Thus, we can think of the banks as being arranged along a line with\n",
+ "loans from bank $ i $ being immediately deposited in $ i+1 $\n",
+ "\n",
+ "- in this way, the debtors to bank $ i $ become creditors of\n",
+ " bank $ i+1 $ \n",
+ "\n",
+ "\n",
+ "Finally, we add an *initial condition* about an exogenous level of bank\n",
+ "$ 0 $’s deposits\n",
+ "\n",
+ "$$\n",
+ "D_0 \\ \\text{ is given exogenously}\n",
+ "$$\n",
+ "\n",
+ "We can think of $ D_0 $ as being the amount of cash that a first\n",
+ "depositor put into the first bank in the system, bank number $ i=0 $.\n",
+ "\n",
+ "Now we do a little algebra.\n",
+ "\n",
+ "Combining equations [(10.2)](#equation-balance) and [(10.3)](#equation-reserves) tells us that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "L_i = (1-r) D_i \\tag{10.5}\n",
+ "$$\n",
+ "\n",
+ "This states that bank $ i $ loans a fraction $ (1-r) $ of its\n",
+ "deposits and keeps a fraction $ r $ as cash reserves.\n",
+ "\n",
+ "Combining equation [(10.5)](#equation-fraction) with equation [(10.4)](#equation-deposits) tells us that\n",
+ "\n",
+ "$$\n",
+ "D_{i+1} = (1-r) D_i \\ \\text{ for } i \\geq 0\n",
+ "$$\n",
+ "\n",
+ "which implies that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "D_i = (1 - r)^i D_0 \\ \\text{ for } i \\geq 0 \\tag{10.6}\n",
+ "$$\n",
+ "\n",
+ "Equation [(10.6)](#equation-geomseries) expresses $ D_i $ as the $ i $ th term in the\n",
+ "product of $ D_0 $ and the geometric series\n",
+ "\n",
+ "$$\n",
+ "1, (1-r), (1-r)^2, \\cdots\n",
+ "$$\n",
+ "\n",
+ "Therefore, the sum of all deposits in our banking system\n",
+ "$ i=0, 1, 2, \\ldots $ is\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\sum_{i=0}^\\infty (1-r)^i D_0 = \\frac{D_0}{1 - (1-r)} = \\frac{D_0}{r} \\tag{10.7}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a01a1044",
+ "metadata": {},
+ "source": [
+ "### Money multiplier\n",
+ "\n",
+ "The **money multiplier** is a number that tells the multiplicative\n",
+ "factor by which an exogenous injection of cash into bank $ 0 $ leads\n",
+ "to an increase in the total deposits in the banking system.\n",
+ "\n",
+ "Equation [(10.7)](#equation-sumdeposits) asserts that the **money multiplier** is\n",
+ "$ \\frac{1}{r} $\n",
+ "\n",
+ "- An initial deposit of cash of $ D_0 $ in bank $ 0 $ leads\n",
+ " the banking system to create total deposits of $ \\frac{D_0}{r} $. \n",
+ "- The initial deposit $ D_0 $ is held as reserves, distributed\n",
+ " throughout the banking system according to $ D_0 = \\sum_{i=0}^\\infty R_i $. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "24ee3073",
+ "metadata": {},
+ "source": [
+ "## Example: The Keynesian Multiplier\n",
+ "\n",
+ "The famous economist John Maynard Keynes and his followers created a\n",
+ "simple model intended to determine national income $ y $ in\n",
+ "circumstances in which\n",
+ "\n",
+ "- there are substantial unemployed resources, in particular **excess\n",
+ " supply** of labor and capital \n",
+ "- prices and interest rates fail to adjust to make aggregate **supply\n",
+ " equal demand** (e.g., prices and interest rates are frozen) \n",
+ "- national income is entirely determined by aggregate demand "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "924a8290",
+ "metadata": {},
+ "source": [
+ "### Static version\n",
+ "\n",
+ "An elementary Keynesian model of national income determination consists\n",
+ "of three equations that describe aggregate demand for $ y $ and its\n",
+ "components.\n",
+ "\n",
+ "The first equation is a national income identity asserting that\n",
+ "consumption $ c $ plus investment $ i $ equals national income\n",
+ "$ y $:\n",
+ "\n",
+ "$$\n",
+ "c+ i = y\n",
+ "$$\n",
+ "\n",
+ "The second equation is a Keynesian consumption function asserting that\n",
+ "people consume a fraction $ b \\in (0,1) $ of their income:\n",
+ "\n",
+ "$$\n",
+ "c = b y\n",
+ "$$\n",
+ "\n",
+ "The fraction $ b \\in (0,1) $ is called the **marginal propensity to\n",
+ "consume**.\n",
+ "\n",
+ "The fraction $ 1-b \\in (0,1) $ is called the **marginal propensity\n",
+ "to save**.\n",
+ "\n",
+ "The third equation simply states that investment is exogenous at level\n",
+ "$ i $.\n",
+ "\n",
+ "- *exogenous* means *determined outside this model*. \n",
+ "\n",
+ "\n",
+ "Substituting the second equation into the first gives $ (1-b) y = i $.\n",
+ "\n",
+ "Solving this equation for $ y $ gives\n",
+ "\n",
+ "$$\n",
+ "y = \\frac{1}{1-b} i\n",
+ "$$\n",
+ "\n",
+ "The quantity $ \\frac{1}{1-b} $ is called the **investment\n",
+ "multiplier** or simply the **multiplier**.\n",
+ "\n",
+ "Applying the formula for the sum of an infinite geometric series, we can\n",
+ "write the above equation as\n",
+ "\n",
+ "$$\n",
+ "y = i \\sum_{t=0}^\\infty b^t\n",
+ "$$\n",
+ "\n",
+ "where $ t $ is a nonnegative integer.\n",
+ "\n",
+ "So we arrive at the following equivalent expressions for the multiplier:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{1-b} = \\sum_{t=0}^\\infty b^t\n",
+ "$$\n",
+ "\n",
+ "The expression $ \\sum_{t=0}^\\infty b^t $ motivates an interpretation\n",
+ "of the multiplier as the outcome of a dynamic process that we describe\n",
+ "next."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d5bb685f",
+ "metadata": {},
+ "source": [
+ "### Dynamic version\n",
+ "\n",
+ "We arrive at a dynamic version by interpreting the nonnegative integer\n",
+ "$ t $ as indexing time and changing our specification of the\n",
+ "consumption function to take time into account\n",
+ "\n",
+ "- we add a one-period lag in how income affects consumption \n",
+ "\n",
+ "\n",
+ "We let $ c_t $ be consumption at time $ t $ and $ i_t $ be\n",
+ "investment at time $ t $.\n",
+ "\n",
+ "We modify our consumption function to assume the form\n",
+ "\n",
+ "$$\n",
+ "c_t = b y_{t-1}\n",
+ "$$\n",
+ "\n",
+ "so that $ b $ is the marginal propensity to consume (now) out of\n",
+ "last period’s income.\n",
+ "\n",
+ "We begin with an initial condition stating that\n",
+ "\n",
+ "$$\n",
+ "y_{-1} = 0\n",
+ "$$\n",
+ "\n",
+ "We also assume that\n",
+ "\n",
+ "$$\n",
+ "i_t = i \\ \\ \\textrm {for all } t \\geq 0\n",
+ "$$\n",
+ "\n",
+ "so that investment is constant over time.\n",
+ "\n",
+ "It follows that\n",
+ "\n",
+ "$$\n",
+ "y_0 = i + c_0 = i + b y_{-1} = i\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "$$\n",
+ "y_1 = c_1 + i = b y_0 + i = (1 + b) i\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "$$\n",
+ "y_2 = c_2 + i = b y_1 + i = (1 + b + b^2) i\n",
+ "$$\n",
+ "\n",
+ "and more generally\n",
+ "\n",
+ "$$\n",
+ "y_t = b y_{t-1} + i = (1+ b + b^2 + \\cdots + b^t) i\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "y_t = \\frac{1-b^{t+1}}{1 -b } i\n",
+ "$$\n",
+ "\n",
+ "Evidently, as $ t \\rightarrow + \\infty $,\n",
+ "\n",
+ "$$\n",
+ "y_t \\rightarrow \\frac{1}{1-b} i\n",
+ "$$\n",
+ "\n",
+ "**Remark 1:** The above formula is often applied to assert that an\n",
+ "exogenous increase in investment of $ \\Delta i $ at time $ 0 $\n",
+ "ignites a dynamic process of increases in national income by successive amounts\n",
+ "\n",
+ "$$\n",
+ "\\Delta i, (1 + b )\\Delta i, (1+b + b^2) \\Delta i , \\cdots\n",
+ "$$\n",
+ "\n",
+ "at times $ 0, 1, 2, \\ldots $.\n",
+ "\n",
+ "**Remark 2** Let $ g_t $ be an exogenous sequence of government\n",
+ "expenditures.\n",
+ "\n",
+ "If we generalize the model so that the national income identity\n",
+ "becomes\n",
+ "\n",
+ "$$\n",
+ "c_t + i_t + g_t = y_t\n",
+ "$$\n",
+ "\n",
+ "then a version of the preceding argument shows that the **government\n",
+ "expenditures multiplier** is also $ \\frac{1}{1-b} $, so that a\n",
+ "permanent increase in government expenditures ultimately leads to an\n",
+ "increase in national income equal to the multiplier times the increase\n",
+ "in government expenditures."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e6df577c",
+ "metadata": {},
+ "source": [
+ "## Example: Interest Rates and Present Values\n",
+ "\n",
+ "We can apply our formula for geometric series to study how interest\n",
+ "rates affect values of streams of dollar payments that extend over time.\n",
+ "\n",
+ "We work in discrete time and assume that $ t = 0, 1, 2, \\ldots $\n",
+ "indexes time.\n",
+ "\n",
+ "We let $ r \\in (0,1) $ be a one-period **net nominal interest rate**\n",
+ "\n",
+ "- if the nominal interest rate is $ 5 $ percent,\n",
+ " then $ r= .05 $ \n",
+ "\n",
+ "\n",
+ "A one-period **gross nominal interest rate** $ R $ is defined as\n",
+ "\n",
+ "$$\n",
+ "R = 1 + r \\in (1, 2)\n",
+ "$$\n",
+ "\n",
+ "- if $ r=.05 $, then $ R = 1.05 $ \n",
+ "\n",
+ "\n",
+ "**Remark:** The gross nominal interest rate $ R $ is an **exchange\n",
+ "rate** or **relative price** of dollars at between times $ t $ and\n",
+ "$ t+1 $. The units of $ R $ are dollars at time $ t+1 $ per\n",
+ "dollar at time $ t $.\n",
+ "\n",
+ "When people borrow and lend, they trade dollars now for dollars later or\n",
+ "dollars later for dollars now.\n",
+ "\n",
+ "The price at which these exchanges occur is the gross nominal interest\n",
+ "rate.\n",
+ "\n",
+ "- If I sell $ x $ dollars to you today, you pay me $ R x $\n",
+ " dollars tomorrow. \n",
+ "- This means that you borrowed $ x $ dollars for me at a gross\n",
+ " interest rate $ R $ and a net interest rate $ r $. \n",
+ "\n",
+ "\n",
+ "We assume that the net nominal interest rate $ r $ is fixed over\n",
+ "time, so that $ R $ is the gross nominal interest rate at times\n",
+ "$ t=0, 1, 2, \\ldots $.\n",
+ "\n",
+ "Two important geometric sequences are\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "1, R, R^2, \\cdots \\tag{10.8}\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "1, R^{-1}, R^{-2}, \\cdots \\tag{10.9}\n",
+ "$$\n",
+ "\n",
+ "Sequence [(10.8)](#equation-geom1) tells us how dollar values of an investment **accumulate**\n",
+ "through time.\n",
+ "\n",
+ "Sequence [(10.9)](#equation-geom2) tells us how to **discount** future dollars to get their\n",
+ "values in terms of today’s dollars."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "55077a70",
+ "metadata": {},
+ "source": [
+ "### Accumulation\n",
+ "\n",
+ "Geometric sequence [(10.8)](#equation-geom1) tells us how one dollar invested and re-invested\n",
+ "in a project with gross one period nominal rate of return accumulates\n",
+ "\n",
+ "- here we assume that net interest payments are reinvested in the\n",
+ " project \n",
+ "- thus, $ 1 $ dollar invested at time $ 0 $ pays interest\n",
+ " $ r $ dollars after one period, so we have $ r+1 = R $\n",
+ " dollars at time$ 1 $ \n",
+ "- at time $ 1 $ we reinvest $ 1+r =R $ dollars and receive interest\n",
+ " of $ r R $ dollars at time $ 2 $ plus the *principal*\n",
+ " $ R $ dollars, so we receive $ r R + R = (1+r)R = R^2 $\n",
+ " dollars at the end of period $ 2 $ \n",
+ "- and so on \n",
+ "\n",
+ "\n",
+ "Evidently, if we invest $ x $ dollars at time $ 0 $ and\n",
+ "reinvest the proceeds, then the sequence\n",
+ "\n",
+ "$$\n",
+ "x , xR , x R^2, \\cdots\n",
+ "$$\n",
+ "\n",
+ "tells how our account accumulates at dates $ t=0, 1, 2, \\ldots $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "09c12119",
+ "metadata": {},
+ "source": [
+ "### Discounting\n",
+ "\n",
+ "Geometric sequence [(10.9)](#equation-geom2) tells us how much future dollars are worth in terms of today’s dollars.\n",
+ "\n",
+ "Remember that the units of $ R $ are dollars at $ t+1 $ per\n",
+ "dollar at $ t $.\n",
+ "\n",
+ "It follows that\n",
+ "\n",
+ "- the units of $ R^{-1} $ are dollars at $ t $ per dollar at $ t+1 $ \n",
+ "- the units of $ R^{-2} $ are dollars at $ t $ per dollar at $ t+2 $ \n",
+ "- and so on; the units of $ R^{-j} $ are dollars at $ t $ per\n",
+ " dollar at $ t+j $ \n",
+ "\n",
+ "\n",
+ "So if someone has a claim on $ x $ dollars at time $ t+j $, it\n",
+ "is worth $ x R^{-j} $ dollars at time $ t $ (e.g., today)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "09aa5f66",
+ "metadata": {},
+ "source": [
+ "### Application to asset pricing\n",
+ "\n",
+ "A **lease** requires a payments stream of $ x_t $ dollars at\n",
+ "times $ t = 0, 1, 2, \\ldots $ where\n",
+ "\n",
+ "$$\n",
+ "x_t = G^t x_0\n",
+ "$$\n",
+ "\n",
+ "where $ G = (1+g) $ and $ g \\in (0,1) $.\n",
+ "\n",
+ "Thus, lease payments increase at $ g $ percent per period.\n",
+ "\n",
+ "For a reason soon to be revealed, we assume that $ G < R $.\n",
+ "\n",
+ "The **present value** of the lease is\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned} p_0 & = x_0 + x_1/R + x_2/(R^2) + \\cdots \\\\\n",
+ " & = x_0 (1 + G R^{-1} + G^2 R^{-2} + \\cdots ) \\\\\n",
+ " & = x_0 \\frac{1}{1 - G R^{-1}} \\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "where the last line uses the formula for an infinite geometric series.\n",
+ "\n",
+ "Recall that $ R = 1+r $ and $ G = 1+g $ and that $ R > G $\n",
+ "and $ r > g $ and that $ r $ and $ g $ are typically small\n",
+ "numbers, e.g., .05 or .03.\n",
+ "\n",
+ "Use the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) of $ \\frac{1}{1+r} $ about $ r=0 $,\n",
+ "namely,\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{1+r} = 1 - r + r^2 - r^3 + \\cdots\n",
+ "$$\n",
+ "\n",
+ "and the fact that $ r $ is small to approximate\n",
+ "$ \\frac{1}{1+r} \\approx 1 - r $.\n",
+ "\n",
+ "Use this approximation to write $ p_0 $ as\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ " p_0 &= x_0 \\frac{1}{1 - G R^{-1}} \\\\\n",
+ " &= x_0 \\frac{1}{1 - (1+g) (1-r) } \\\\\n",
+ " &= x_0 \\frac{1}{1 - (1+g - r - rg)} \\\\\n",
+ " & \\approx x_0 \\frac{1}{r -g }\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "where the last step uses the approximation $ r g \\approx 0 $.\n",
+ "\n",
+ "The approximation\n",
+ "\n",
+ "$$\n",
+ "p_0 = \\frac{x_0 }{r -g }\n",
+ "$$\n",
+ "\n",
+ "is known as the **Gordon formula** for the present value or current\n",
+ "price of an infinite payment stream $ x_0 G^t $ when the nominal\n",
+ "one-period interest rate is $ r $ and when $ r > g $.\n",
+ "\n",
+ "We can also extend the asset pricing formula so that it applies to finite leases.\n",
+ "\n",
+ "Let the payment stream on the lease now be $ x_t $ for $ t= 1,2, \\dots,T $, where again\n",
+ "\n",
+ "$$\n",
+ "x_t = G^t x_0\n",
+ "$$\n",
+ "\n",
+ "The present value of this lease is:\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned} \\begin{split}p_0&=x_0 + x_1/R + \\dots +x_T/R^T \\\\ &= x_0(1+GR^{-1}+\\dots +G^{T}R^{-T}) \\\\ &= \\frac{x_0(1-G^{T+1}R^{-(T+1)})}{1-GR^{-1}} \\end{split}\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "Applying the Taylor series to $ R^{-(T+1)} $ about $ r=0 $ we get:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{(1+r)^{T+1}}= 1-r(T+1)+\\frac{1}{2}r^2(T+1)(T+2)+\\dots \\approx 1-r(T+1)\n",
+ "$$\n",
+ "\n",
+ "Similarly, applying the Taylor series to $ G^{T+1} $ about $ g=0 $:\n",
+ "\n",
+ "$$\n",
+ "(1+g)^{T+1} = 1+(T+1)g+\\frac{T(T+1)}{2!}g^2+\\frac{(T-1)T(T+1)}{3!}g^3+\\dots \\approx 1+ (T+1)g\n",
+ "$$\n",
+ "\n",
+ "Thus, we get the following approximation:\n",
+ "\n",
+ "$$\n",
+ "p_0 =\\frac{x_0(1-(1+(T+1)g)(1-r(T+1)))}{1-(1-r)(1+g) }\n",
+ "$$\n",
+ "\n",
+ "Expanding:\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned} p_0 &=\\frac{x_0(1-1+(T+1)^2 rg +r(T+1)-g(T+1))}{1-1+r-g+rg} \\\\&=\\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\\\ &= \\frac{x_0(T+1)(r-g)}{r-g + rg}+\\frac{x_0rg(T+1)^2}{r-g+rg}\\\\ &\\approx \\frac{x_0(T+1)(r-g)}{r-g}+\\frac{x_0rg(T+1)}{r-g}\\\\ &= x_0(T+1) + \\frac{x_0rg(T+1)}{r-g} \\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "We could have also approximated by removing the second term\n",
+ "$ rgx_0(T+1) $ when $ T $ is relatively small compared to\n",
+ "$ 1/(rg) $ to get $ x_0(T+1) $ as in the finite stream\n",
+ "approximation.\n",
+ "\n",
+ "We will plot the true finite stream present-value and the two\n",
+ "approximations, under different values of $ T $, and $ g $ and $ r $ in Python.\n",
+ "\n",
+ "First we plot the true finite stream present-value after computing it\n",
+ "below"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "464798aa",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# True present value of a finite lease\n",
+ "def finite_lease_pv_true(T, g, r, x_0):\n",
+ " G = (1 + g)\n",
+ " R = (1 + r)\n",
+ " return (x_0 * (1 - G**(T + 1) * R**(-T - 1))) / (1 - G * R**(-1))\n",
+ "# First approximation for our finite lease\n",
+ "\n",
+ "def finite_lease_pv_approx_1(T, g, r, x_0):\n",
+ " p = x_0 * (T + 1) + x_0 * r * g * (T + 1) / (r - g)\n",
+ " return p\n",
+ "\n",
+ "# Second approximation for our finite lease\n",
+ "def finite_lease_pv_approx_2(T, g, r, x_0):\n",
+ " return (x_0 * (T + 1))\n",
+ "\n",
+ "# Infinite lease\n",
+ "def infinite_lease(g, r, x_0):\n",
+ " G = (1 + g)\n",
+ " R = (1 + r)\n",
+ " return x_0 / (1 - G * R**(-1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "317b20a3",
+ "metadata": {},
+ "source": [
+ "Now that we have defined our functions, we can plot some outcomes.\n",
+ "\n",
+ "First we study the quality of our approximations"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a9442900",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def plot_function(axes, x_vals, func, args):\n",
+ " axes.plot(x_vals, func(*args), label=func.__name__)\n",
+ "\n",
+ "T_max = 50\n",
+ "\n",
+ "T = np.arange(0, T_max+1)\n",
+ "g = 0.02\n",
+ "r = 0.03\n",
+ "x_0 = 1\n",
+ "\n",
+ "our_args = (T, g, r, x_0)\n",
+ "funcs = [finite_lease_pv_true,\n",
+ " finite_lease_pv_approx_1,\n",
+ " finite_lease_pv_approx_2]\n",
+ " # the three functions we want to compare\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "for f in funcs:\n",
+ " plot_function(ax, T, f, our_args)\n",
+ "ax.legend()\n",
+ "ax.set_xlabel('$T$ Periods Ahead')\n",
+ "ax.set_ylabel('Present Value, $p_0$')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "68e7c165",
+ "metadata": {},
+ "source": [
+ "Evidently our approximations perform well for small values of $ T $.\n",
+ "\n",
+ "However, holding $ g $ and r fixed, our approximations deteriorate as $ T $ increases.\n",
+ "\n",
+ "Next we compare the infinite and finite duration lease present values\n",
+ "over different lease lengths $ T $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "35ab4356",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Convergence of infinite and finite\n",
+ "T_max = 1000\n",
+ "T = np.arange(0, T_max+1)\n",
+ "fig, ax = plt.subplots()\n",
+ "f_1 = finite_lease_pv_true(T, g, r, x_0)\n",
+ "f_2 = np.full(T_max+1, infinite_lease(g, r, x_0))\n",
+ "ax.plot(T, f_1, label='T-period lease PV')\n",
+ "ax.plot(T, f_2, '--', label='Infinite lease PV')\n",
+ "ax.set_xlabel('$T$ Periods Ahead')\n",
+ "ax.set_ylabel('Present Value, $p_0$')\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cab245c2",
+ "metadata": {},
+ "source": [
+ "The graph above shows how as duration $ T \\rightarrow +\\infty $,\n",
+ "the value of a lease of duration $ T $ approaches the value of a\n",
+ "perpetual lease.\n",
+ "\n",
+ "Now we consider two different views of what happens as $ r $ and\n",
+ "$ g $ covary"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "850420a3",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# First view\n",
+ "# Changing r and g\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.set_ylabel('Present Value, $p_0$')\n",
+ "ax.set_xlabel('$T$ periods ahead')\n",
+ "T_max = 10\n",
+ "T=np.arange(0, T_max+1)\n",
+ "\n",
+ "rs, gs = (0.9, 0.5, 0.4001, 0.4), (0.4, 0.4, 0.4, 0.5),\n",
+ "comparisons = (r'$\\gg$', '$>$', r'$\\approx$', '$<$')\n",
+ "for r, g, comp in zip(rs, gs, comparisons):\n",
+ " ax.plot(finite_lease_pv_true(T, g, r, x_0), label=f'r(={r}) {comp} g(={g})')\n",
+ "\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "58cec78f",
+ "metadata": {},
+ "source": [
+ "This graph gives a big hint for why the condition $ r > g $ is\n",
+ "necessary if a lease of length $ T = +\\infty $ is to have finite\n",
+ "value.\n",
+ "\n",
+ "For fans of 3-d graphs the same point comes through in the following\n",
+ "graph.\n",
+ "\n",
+ "If you aren’t enamored of 3-d graphs, feel free to skip the next\n",
+ "visualization!"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e7e0f541",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Second view\n",
+ "fig = plt.figure(figsize = [16, 5])\n",
+ "T = 3\n",
+ "ax = plt.subplot(projection='3d')\n",
+ "r = np.arange(0.01, 0.99, 0.005)\n",
+ "g = np.arange(0.011, 0.991, 0.005)\n",
+ "\n",
+ "rr, gg = np.meshgrid(r, g)\n",
+ "z = finite_lease_pv_true(T, gg, rr, x_0)\n",
+ "\n",
+ "# Removes points where undefined\n",
+ "same = (rr == gg)\n",
+ "z[same] = np.nan\n",
+ "surf = ax.plot_surface(rr, gg, z, cmap=cm.coolwarm,\n",
+ " antialiased=True, clim=(0, 15))\n",
+ "fig.colorbar(surf, shrink=0.5, aspect=5)\n",
+ "ax.set_xlabel('$r$')\n",
+ "ax.set_ylabel('$g$')\n",
+ "ax.set_zlabel('Present Value, $p_0$')\n",
+ "ax.view_init(20, 8)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0f7994f1",
+ "metadata": {},
+ "source": [
+ "We can use a little calculus to study how the present value $ p_0 $\n",
+ "of a lease varies with $ r $ and $ g $.\n",
+ "\n",
+ "We will use a library called [SymPy](https://www.sympy.org/).\n",
+ "\n",
+ "SymPy enables us to do symbolic math calculations including\n",
+ "computing derivatives of algebraic equations.\n",
+ "\n",
+ "We will illustrate how it works by creating a symbolic expression that\n",
+ "represents our present value formula for an infinite lease.\n",
+ "\n",
+ "After that, we’ll use SymPy to compute derivatives"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e1dc156f",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Creates algebraic symbols that can be used in an algebraic expression\n",
+ "g, r, x0 = sym.symbols('g, r, x0')\n",
+ "G = (1 + g)\n",
+ "R = (1 + r)\n",
+ "p0 = x0 / (1 - G * R**(-1))\n",
+ "init_printing(use_latex='mathjax')\n",
+ "print('Our formula is:')\n",
+ "p0"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c224172e",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "print('dp0 / dg is:')\n",
+ "dp_dg = sym.diff(p0, g)\n",
+ "dp_dg"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "bdad22d4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "print('dp0 / dr is:')\n",
+ "dp_dr = sym.diff(p0, r)\n",
+ "dp_dr"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3f73bfce",
+ "metadata": {},
+ "source": [
+ "We can see that for $ \\frac{\\partial p_0}{\\partial r}<0 $ as long as\n",
+ "$ r>g $, $ r>0 $ and $ g>0 $ and $ x_0 $ is positive,\n",
+ "so $ \\frac{\\partial p_0}{\\partial r} $ will always be negative.\n",
+ "\n",
+ "Similarly, $ \\frac{\\partial p_0}{\\partial g}>0 $ as long as $ r>g $, $ r>0 $ and $ g>0 $ and $ x_0 $ is positive, so $ \\frac{\\partial p_0}{\\partial g} $\n",
+ "will always be positive."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "42684d97",
+ "metadata": {},
+ "source": [
+ "## Back to the Keynesian multiplier\n",
+ "\n",
+ "We will now go back to the case of the Keynesian multiplier and plot the\n",
+ "time path of $ y_t $, given that consumption is a constant fraction\n",
+ "of national income, and investment is fixed."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "77993f1f",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Function that calculates a path of y\n",
+ "def calculate_y(i, b, g, T, y_init):\n",
+ " y = np.zeros(T+1)\n",
+ " y[0] = i + b * y_init + g\n",
+ " for t in range(1, T+1):\n",
+ " y[t] = b * y[t-1] + i + g\n",
+ " return y\n",
+ "\n",
+ "# Initial values\n",
+ "i_0 = 0.3\n",
+ "g_0 = 0.3\n",
+ "# 2/3 of income goes towards consumption\n",
+ "b = 2/3\n",
+ "y_init = 0\n",
+ "T = 100\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.set_xlabel('$t$')\n",
+ "ax.set_ylabel('$y_t$')\n",
+ "ax.plot(np.arange(0, T+1), calculate_y(i_0, b, g_0, T, y_init))\n",
+ "# Output predicted by geometric series\n",
+ "ax.hlines(i_0 / (1 - b) + g_0 / (1 - b), xmin=-1, xmax=101, linestyles='--')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d60ce1dd",
+ "metadata": {},
+ "source": [
+ "In this model, income grows over time, until it gradually converges to\n",
+ "the infinite geometric series sum of income.\n",
+ "\n",
+ "We now examine what will\n",
+ "happen if we vary the so-called **marginal propensity to consume**,\n",
+ "i.e., the fraction of income that is consumed"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "19c7967c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "bs = (1/3, 2/3, 5/6, 0.9)\n",
+ "\n",
+ "fig,ax = plt.subplots()\n",
+ "ax.set_ylabel('$y_t$')\n",
+ "ax.set_xlabel('$t$')\n",
+ "x = np.arange(0, T+1)\n",
+ "for b in bs:\n",
+ " y = calculate_y(i_0, b, g_0, T, y_init)\n",
+ " ax.plot(x, y, label=r'$b=$'+f\"{b:.2f}\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0482576a",
+ "metadata": {},
+ "source": [
+ "Increasing the marginal propensity to consume $ b $ increases the\n",
+ "path of output over time.\n",
+ "\n",
+ "Now we will compare the effects on output of increases in investment and government spending."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "69993f96",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 10))\n",
+ "fig.subplots_adjust(hspace=0.3)\n",
+ "\n",
+ "x = np.arange(0, T+1)\n",
+ "values = [0.3, 0.4]\n",
+ "\n",
+ "for i in values:\n",
+ " y = calculate_y(i, b, g_0, T, y_init)\n",
+ " ax1.plot(x, y, label=f\"i={i}\")\n",
+ "for g in values:\n",
+ " y = calculate_y(i_0, b, g, T, y_init)\n",
+ " ax2.plot(x, y, label=f\"g={g}\")\n",
+ "\n",
+ "axes = ax1, ax2\n",
+ "param_labels = \"Investment\", \"Government Spending\"\n",
+ "for ax, param in zip(axes, param_labels):\n",
+ " ax.set_title(f'An Increase in {param} on Output')\n",
+ " ax.legend(loc =\"lower right\")\n",
+ " ax.set_ylabel('$y_t$')\n",
+ " ax.set_xlabel('$t$')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "18f1323b",
+ "metadata": {},
+ "source": [
+ "Notice here, whether government spending increases from 0.3 to 0.4 or\n",
+ "investment increases from 0.3 to 0.4, the shifts in the graphs are\n",
+ "identical."
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476280.879984,
+ "filename": "geom_series.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Geometric Series for Elementary Economics"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/greek_square.ipynb b/_notebooks/greek_square.ipynb
new file mode 100644
index 000000000..ac09f8a85
--- /dev/null
+++ b/_notebooks/greek_square.ipynb
@@ -0,0 +1,1033 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "7721661a",
+ "metadata": {},
+ "source": [
+ "# Computing Square Roots"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a817be8f",
+ "metadata": {},
+ "source": [
+ "## Introduction\n",
+ "\n",
+ "Chapter 24 of [[Russell, 2004](https://intro.quantecon.org/zreferences.html#id3)] about early Greek mathematics and astronomy contains this\n",
+ "fascinating passage:\n",
+ "\n",
+ "> The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value were discovered. The best was as follows: Form two columns of numbers, which we will call the $ a $’s and the $ b $’s; each starts with a $ 1 $. The next $ a $, at each stage, is formed by adding the last $ a $ and the $ b $ already obtained; the next $ b $ is formed by adding twice the previous $ a $ to the previous $ b $. The first 6 pairs so obtained are $ (1,1), (2,3), (5,7), (12,17), (29,41), (70,99) $. In each pair, $ 2 a^2 - b^2 $ is $ 1 $ or $ -1 $. Thus $ b/a $ is nearly the square root of two, and at each fresh step it gets nearer. For instance, the reader may satisy himself that the square of $ 99/70 $ is very nearly equal to $ 2 $.\n",
+ "\n",
+ "\n",
+ "This lecture drills down and studies this ancient method for computing square roots by using some of the matrix algebra that we’ve learned in earlier quantecon lectures.\n",
+ "\n",
+ "In particular, this lecture can be viewed as a sequel to [Eigenvalues and Eigenvectors](https://intro.quantecon.org/eigen_I.html).\n",
+ "\n",
+ "It provides an example of how eigenvectors isolate *invariant subspaces* that help construct and analyze solutions of linear difference equations.\n",
+ "\n",
+ "When vector $ x_t $ starts in an invariant subspace, iterating the different equation keeps $ x_{t+j} $\n",
+ "in that subspace for all $ j \\geq 1 $.\n",
+ "\n",
+ "Invariant subspace methods are used throughout applied economic dynamics, for example, in the lecture [Money Financed Government Deficits and Price Levels](https://intro.quantecon.org/money_inflation.html).\n",
+ "\n",
+ "Our approach here is to illustrate the method with an ancient example, one that ancient Greek mathematicians used to compute square roots of positive integers."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cef049bb",
+ "metadata": {},
+ "source": [
+ "## Perfect squares and irrational numbers\n",
+ "\n",
+ "An integer is called a **perfect square** if its square root is also an integer.\n",
+ "\n",
+ "An ordered sequence of perfect squares starts with\n",
+ "\n",
+ "$$\n",
+ "4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, \\ldots\n",
+ "$$\n",
+ "\n",
+ "If an integer is not a perfect square, then its square root is an irrational number – i.e., it cannot be expressed as a ratio of two integers, and its decimal expansion is indefinite.\n",
+ "\n",
+ "The ancient Greeks invented an algorithm to compute square roots of integers, including integers that are not perfect squares.\n",
+ "\n",
+ "Their method involved\n",
+ "\n",
+ "- computing a particular sequence of integers $ \\{y_t\\}_{t=0}^\\infty $; \n",
+ "- computing $ \\lim_{t \\rightarrow \\infty} \\left(\\frac{y_{t+1}}{y_t}\\right) = \\bar r $; \n",
+ "- deducing the desired square root from $ \\bar r $. \n",
+ "\n",
+ "\n",
+ "In this lecture, we’ll describe this method.\n",
+ "\n",
+ "We’ll also use invariant subspaces to describe variations on this method that are faster."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e121dc50",
+ "metadata": {},
+ "source": [
+ "## Second-order linear difference equations\n",
+ "\n",
+ "Before telling how the ancient Greeks computed square roots, we’ll provide a quick introduction\n",
+ "to second-order linear difference equations.\n",
+ "\n",
+ "We’ll study the following second-order linear difference equation\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_t = a_1 y_{t-1} + a_2 y_{t-2}, \\quad t \\geq 0 \\tag{18.1}\n",
+ "$$\n",
+ "\n",
+ "where $ (y_{-1}, y_{-2}) $ is a pair of given initial conditions.\n",
+ "\n",
+ "Equation [(18.1)](#equation-eq-2diff1) is actually an infinite number of linear equations in the sequence\n",
+ "$ \\{y_t\\}_{t=0}^\\infty $.\n",
+ "\n",
+ "There is one equation each for $ t = 0, 1, 2, \\ldots $.\n",
+ "\n",
+ "We could follow an approach taken in the lecture on [present values](https://intro.quantecon.org/pv.html) and stack all of these equations into a single matrix equation that we would then solve by using matrix inversion.\n",
+ "\n",
+ ">**Note**\n",
+ ">\n",
+ ">In the present instance, the matrix equation would multiply a countably infinite dimensional square matrix by a countably infinite dimensional vector. With some qualifications, matrix multiplication and inversion tools apply to such an equation.\n",
+ "\n",
+ "But we won’t pursue that approach here.\n",
+ "\n",
+ "Instead, we’ll seek to find a time-invariant function that *solves* our difference equation, meaning\n",
+ "that it provides a formula for a $ \\{y_t\\}_{t=0}^\\infty $ sequence that satisfies\n",
+ "equation [(18.1)](#equation-eq-2diff1) for each $ t \\geq 0 $.\n",
+ "\n",
+ "We seek an expression for $ y_t, t \\geq 0 $ as functions of the initial conditions $ (y_{-1}, y_{-2}) $:\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_t = g((y_{-1}, y_{-2});t), \\quad t \\geq 0. \\tag{18.2}\n",
+ "$$\n",
+ "\n",
+ "We call such a function $ g $ a *solution* of the difference equation [(18.1)](#equation-eq-2diff1).\n",
+ "\n",
+ "One way to discover a solution is to use a guess and verify method.\n",
+ "\n",
+ "We shall begin by considering a special initial pair of initial conditions\n",
+ "that satisfy\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_{-1} = \\delta y_{-2} \\tag{18.3}\n",
+ "$$\n",
+ "\n",
+ "where $ \\delta $ is a scalar to be determined.\n",
+ "\n",
+ "For initial condition that satisfy [(18.3)](#equation-eq-2diff3)\n",
+ "equation [(18.1)](#equation-eq-2diff1) impllies that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_0 = \\left(a_1 + \\frac{a_2}{\\delta}\\right) y_{-1}. \\tag{18.4}\n",
+ "$$\n",
+ "\n",
+ "We want\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\left(a_1 + \\frac{a_2}{\\delta}\\right) = \\delta \\tag{18.5}\n",
+ "$$\n",
+ "\n",
+ "which we can rewrite as the *characteristic equation*\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\delta^2 - a_1 \\delta - a_2 = 0. \\tag{18.6}\n",
+ "$$\n",
+ "\n",
+ "Applying the quadratic formula to solve for the roots of [(18.6)](#equation-eq-2diff6) we find that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\delta = \\frac{ a_1 \\pm \\sqrt{a_1^2 + 4 a_2}}{2}. \\tag{18.7}\n",
+ "$$\n",
+ "\n",
+ "For either of the two $ \\delta $’s that satisfy equation [(18.7)](#equation-eq-2diff7),\n",
+ "a solution of difference equation [(18.1)](#equation-eq-2diff1) is\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_t = \\delta^t y_0 , \\forall t \\geq 0 \\tag{18.8}\n",
+ "$$\n",
+ "\n",
+ "provided that we set\n",
+ "\n",
+ "$$\n",
+ "y_0 = \\delta y_{-1} .\n",
+ "$$\n",
+ "\n",
+ "The *general* solution of difference equation [(18.1)](#equation-eq-2diff1) takes the form\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_t = \\eta_1 \\delta_1^t + \\eta_2 \\delta_2^t \\tag{18.9}\n",
+ "$$\n",
+ "\n",
+ "where $ \\delta_1, \\delta_2 $ are the two solutions [(18.7)](#equation-eq-2diff7) of the characteristic equation [(18.6)](#equation-eq-2diff6), and $ \\eta_1, \\eta_2 $ are two constants chosen to satisfy\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix} y_{-1} \\cr y_{-2} \\end{bmatrix} = \\begin{bmatrix} \\delta_1^{-1} & \\delta_2^{-1} \\cr \\delta_1^{-2} & \\delta_2^{-2} \\end{bmatrix} \\begin{bmatrix} \\eta_1 \\cr \\eta_2 \\end{bmatrix} \\tag{18.10}\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix} \\eta_1 \\cr \\eta_2 \\end{bmatrix} = \\begin{bmatrix} \\delta_1^{-1} & \\delta_2^{-1} \\cr \\delta_1^{-2} & \\delta_2^{-2} \\end{bmatrix}^{-1} \\begin{bmatrix} y_{-1} \\cr y_{-2} \\end{bmatrix} \\tag{18.11}\n",
+ "$$\n",
+ "\n",
+ "Sometimes we are free to choose the initial conditions $ (y_{-1}, y_{-2}) $, in which case we\n",
+ "use system [(18.10)](#equation-eq-2diff10) to find the associated $ (\\eta_1, \\eta_2) $.\n",
+ "\n",
+ "If we choose $ (y_{-1}, y_{-2}) $ to set $ (\\eta_1, \\eta_2) = (1, 0) $, then $ y_t = \\delta_1^t $ for all $ t \\geq 0 $.\n",
+ "\n",
+ "If we choose $ (y_{-1}, y_{-2}) $ to set $ (\\eta_1, \\eta_2) = (0, 1) $, then $ y_t = \\delta_2^t $ for all $ t \\geq 0 $.\n",
+ "\n",
+ "Soon we’ll relate the preceding calculations to components an eigen decomposition of a transition matrix that represents difference equation [(18.1)](#equation-eq-2diff1) in a very convenient way.\n",
+ "\n",
+ "We’ll turn to that after we describe how Ancient Greeks figured out how to compute square roots of positive integers that are not perfect squares."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "01b3290a",
+ "metadata": {},
+ "source": [
+ "## Algorithm of the Ancient Greeks\n",
+ "\n",
+ "Let $ \\sigma $ be a positive integer greater than $ 1 $.\n",
+ "\n",
+ "So $ \\sigma \\in {\\mathcal I} \\equiv \\{2, 3, \\ldots \\} $.\n",
+ "\n",
+ "We want an algorithm to compute the square root of $ \\sigma \\in {\\mathcal I} $.\n",
+ "\n",
+ "If $ \\sqrt{\\sigma} \\in {\\mathcal I} $, $ \\sigma $ is said to be a *perfect square*.\n",
+ "\n",
+ "If $ \\sqrt{\\sigma} \\not\\in {\\mathcal I} $, it turns out that it is irrational.\n",
+ "\n",
+ "Ancient Greeks used a recursive algorithm to compute square roots of integers that are not perfect squares.\n",
+ "\n",
+ "The algorithm iterates on a second-order linear difference equation in the sequence $ \\{y_t\\}_{t=0}^\\infty $:\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_{t} = 2 y_{t-1} - (1 - \\sigma) y_{t-2}, \\quad t \\geq 0 \\tag{18.12}\n",
+ "$$\n",
+ "\n",
+ "together with a pair of integers that are initial conditions for $ y_{-1}, y_{-2} $.\n",
+ "\n",
+ "First, we’ll deploy some techniques for solving the difference equations that are also deployed in [Samuelson Multiplier-Accelerator](https://dynamics.quantecon.org/samuelson.html).\n",
+ "\n",
+ "The characteristic equation associated with difference equation [(18.12)](#equation-eq-second-order) is\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "c(x) \\equiv x^2 - 2 x + (1 - \\sigma) = 0 \\tag{18.13}\n",
+ "$$\n",
+ "\n",
+ "(Notice how this is an instance of equation [(18.6)](#equation-eq-2diff6) above.)\n",
+ "\n",
+ "Factoring the right side of equation [(18.13)](#equation-eq-cha-eq0), we obtain\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "c(x)= (x - \\lambda_1) (x-\\lambda_2) = 0 \\tag{18.14}\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "$$\n",
+ "c(x) = 0\n",
+ "$$\n",
+ "\n",
+ "for $ x = \\lambda_1 $ or $ x = \\lambda_2 $.\n",
+ "\n",
+ "These two special values of $ x $ are sometimes called zeros or roots of $ c(x) $.\n",
+ "\n",
+ "By applying the quadratic formula to solve for the roots the characteristic equation\n",
+ "[(18.13)](#equation-eq-cha-eq0), we find that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\lambda_1 = 1 + \\sqrt{\\sigma}, \\quad \\lambda_2 = 1 - \\sqrt{\\sigma}. \\tag{18.15}\n",
+ "$$\n",
+ "\n",
+ "Formulas [(18.15)](#equation-eq-secretweapon) indicate that $ \\lambda_1 $ and $ \\lambda_2 $ are each functions\n",
+ "of a single variable, namely, $ \\sqrt{\\sigma} $, the object that we along with some Ancient Greeks want to compute.\n",
+ "\n",
+ "Ancient Greeks had an indirect way of exploiting this fact to compute square roots of a positive integer.\n",
+ "\n",
+ "They did this by starting from particular initial conditions $ y_{-1}, y_{-2} $ and iterating on the difference equation [(18.12)](#equation-eq-second-order).\n",
+ "\n",
+ "Solutions of difference equation [(18.12)](#equation-eq-second-order) take the form\n",
+ "\n",
+ "$$\n",
+ "y_t = \\lambda_1^t \\eta_1 + \\lambda_2^t \\eta_2\n",
+ "$$\n",
+ "\n",
+ "where $ \\eta_1 $ and $ \\eta_2 $ are chosen to satisfy prescribed initial conditions $ y_{-1}, y_{-2} $:\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "\\lambda_1^{-1} \\eta_1 + \\lambda_2^{-1} \\eta_2 & = y_{-1} \\cr\n",
+ "\\lambda_1^{-2} \\eta_1 + \\lambda_2^{-2} \\eta_2 & = y_{-2}\n",
+ "\\end{aligned} \\tag{18.16}\n",
+ "$$\n",
+ "\n",
+ "System [(18.16)](#equation-eq-leq-sq) of simultaneous linear equations will play a big role in the remainder of this lecture.\n",
+ "\n",
+ "Since $ \\lambda_1 = 1 + \\sqrt{\\sigma} > 1 > \\lambda_2 = 1 - \\sqrt{\\sigma} $,\n",
+ "it follows that for *almost all* (but not all) initial conditions\n",
+ "\n",
+ "$$\n",
+ "\\lim_{t \\rightarrow \\infty} \\left(\\frac{y_{t+1}}{y_t}\\right) = 1 + \\sqrt{\\sigma}.\n",
+ "$$\n",
+ "\n",
+ "Thus,\n",
+ "\n",
+ "$$\n",
+ "\\sqrt{\\sigma} = \\lim_{t \\rightarrow \\infty} \\left(\\frac{y_{t+1}}{y_t}\\right) - 1.\n",
+ "$$\n",
+ "\n",
+ "However, notice that if $ \\eta_1 = 0 $, then\n",
+ "\n",
+ "$$\n",
+ "\\lim_{t \\rightarrow \\infty} \\left(\\frac{y_{t+1}}{y_t}\\right) = 1 - \\sqrt{\\sigma}\n",
+ "$$\n",
+ "\n",
+ "so that\n",
+ "\n",
+ "$$\n",
+ "\\sqrt{\\sigma} = 1 - \\lim_{t \\rightarrow \\infty} \\left(\\frac{y_{t+1}}{y_t}\\right).\n",
+ "$$\n",
+ "\n",
+ "Actually, if $ \\eta_1 =0 $, it follows that\n",
+ "\n",
+ "$$\n",
+ "\\sqrt{\\sigma} = 1 - \\left(\\frac{y_{t+1}}{y_t}\\right) \\quad \\forall t \\geq 0,\n",
+ "$$\n",
+ "\n",
+ "so that convergence is immediate and there is no need to take limits.\n",
+ "\n",
+ "Symmetrically, if $ \\eta_2 =0 $, it follows that\n",
+ "\n",
+ "$$\n",
+ "\\sqrt{\\sigma} = \\left(\\frac{y_{t+1}}{y_t}\\right) - 1 \\quad \\forall t \\geq 0\n",
+ "$$\n",
+ "\n",
+ "so again, convergence is immediate, and we have no need to compute a limit.\n",
+ "\n",
+ "System [(18.16)](#equation-eq-leq-sq) of simultaneous linear equations can be used in various ways.\n",
+ "\n",
+ "- we can take $ y_{-1}, y_{-2} $ as given initial conditions and solve for $ \\eta_1, \\eta_2 $; \n",
+ "- we can instead take $ \\eta_1, \\eta_2 $ as given and solve for initial conditions $ y_{-1}, y_{-2} $. \n",
+ "\n",
+ "\n",
+ "Notice how we used the second approach above when we set $ \\eta_1, \\eta_2 $ either to $ (0, 1) $, for example, or $ (1, 0) $, for example.\n",
+ "\n",
+ "In taking this second approach, we constructed an *invariant subspace* of $ {\\bf R}^2 $.\n",
+ "\n",
+ "Here is what is going on.\n",
+ "\n",
+ "For $ t \\geq 0 $ and for most pairs of initial conditions $ (y_{-1}, y_{-2}) \\in {\\bf R}^2 $ for equation [(18.12)](#equation-eq-second-order), $ y_t $ can be expressed as a linear combination of $ y_{t-1} $ and $ y_{t-2} $.\n",
+ "\n",
+ "But for some special initial conditions $ (y_{-1}, y_{-2}) \\in {\\bf R}^2 $, $ y_t $ can be expressed as a linear function of $ y_{t-1} $ only.\n",
+ "\n",
+ "These special initial conditions require that $ y_{-1} $ be a linear function of $ y_{-2} $.\n",
+ "\n",
+ "We’ll study these special initial conditions soon.\n",
+ "\n",
+ "But first let’s write some Python code to iterate on equation [(18.12)](#equation-eq-second-order) starting from an arbitrary $ (y_{-1}, y_{-2}) \\in {\\bf R}^2 $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "38a1ef00",
+ "metadata": {},
+ "source": [
+ "## Implementation\n",
+ "\n",
+ "We now implement the above algorithm to compute the square root of $ \\sigma $.\n",
+ "\n",
+ "In this lecture, we use the following import:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b94fefbd",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "aefef069",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def solve_λs(coefs): \n",
+ " # Calculate the roots using numpy.roots\n",
+ " λs = np.roots(coefs)\n",
+ " \n",
+ " # Sort the roots for consistency\n",
+ " return sorted(λs, reverse=True)\n",
+ "\n",
+ "def solve_η(λ_1, λ_2, y_neg1, y_neg2):\n",
+ " # Solve the system of linear equation\n",
+ " A = np.array([\n",
+ " [1/λ_1, 1/λ_2],\n",
+ " [1/(λ_1**2), 1/(λ_2**2)]\n",
+ " ])\n",
+ " b = np.array((y_neg1, y_neg2))\n",
+ " ηs = np.linalg.solve(A, b)\n",
+ " \n",
+ " return ηs\n",
+ "\n",
+ "def solve_sqrt(σ, coefs, y_neg1, y_neg2, t_max=100):\n",
+ " # Ensure σ is greater than 1\n",
+ " if σ <= 1:\n",
+ " raise ValueError(\"σ must be greater than 1\")\n",
+ " \n",
+ " # Characteristic roots\n",
+ " λ_1, λ_2 = solve_λs(coefs)\n",
+ " \n",
+ " # Solve for η_1 and η_2\n",
+ " η_1, η_2 = solve_η(λ_1, λ_2, y_neg1, y_neg2)\n",
+ "\n",
+ " # Compute the sequence up to t_max\n",
+ " t = np.arange(t_max + 1)\n",
+ " y = (λ_1 ** t) * η_1 + (λ_2 ** t) * η_2\n",
+ " \n",
+ " # Compute the ratio y_{t+1} / y_t for large t\n",
+ " sqrt_σ_estimate = (y[-1] / y[-2]) - 1\n",
+ " \n",
+ " return sqrt_σ_estimate\n",
+ "\n",
+ "# Use σ = 2 as an example\n",
+ "σ = 2\n",
+ "\n",
+ "# Encode characteristic equation\n",
+ "coefs = (1, -2, (1 - σ))\n",
+ "\n",
+ "# Solve for the square root of σ\n",
+ "sqrt_σ = solve_sqrt(σ, coefs, y_neg1=2, y_neg2=1)\n",
+ "\n",
+ "# Calculate the deviation\n",
+ "dev = abs(sqrt_σ-np.sqrt(σ))\n",
+ "print(f\"sqrt({σ}) is approximately {sqrt_σ:.5f} (error: {dev:.5f})\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6325b52a",
+ "metadata": {},
+ "source": [
+ "Now we consider cases where $ (\\eta_1, \\eta_2) = (0, 1) $ and $ (\\eta_1, \\eta_2) = (1, 0) $"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6097d32b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Compute λ_1, λ_2\n",
+ "λ_1, λ_2 = solve_λs(coefs)\n",
+ "print(f'Roots for the characteristic equation are ({λ_1:.5f}, {λ_2:.5f}))')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c5074ff5",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Case 1: η_1, η_2 = (0, 1)\n",
+ "ηs = (0, 1)\n",
+ "\n",
+ "# Compute y_{t} and y_{t-1} with t >= 0\n",
+ "y = lambda t, ηs: (λ_1 ** t) * ηs[0] + (λ_2 ** t) * ηs[1]\n",
+ "sqrt_σ = 1 - y(1, ηs) / y(0, ηs)\n",
+ "\n",
+ "print(f\"For η_1, η_2 = (0, 1), sqrt_σ = {sqrt_σ:.5f}\")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e6a5214c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Case 2: η_1, η_2 = (1, 0)\n",
+ "ηs = (1, 0)\n",
+ "sqrt_σ = y(1, ηs) / y(0, ηs) - 1\n",
+ "\n",
+ "print(f\"For η_1, η_2 = (1, 0), sqrt_σ = {sqrt_σ:.5f}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0d18e9ba",
+ "metadata": {},
+ "source": [
+ "We find that convergence is immediate.\n",
+ "\n",
+ "Next, we’ll represent the preceding analysis by first vectorizing our second-order difference equation [(18.12)](#equation-eq-second-order) and then using eigendecompositions of an associated state transition matrix."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3d757361",
+ "metadata": {},
+ "source": [
+ "## Vectorizing the difference equation\n",
+ "\n",
+ "Represent [(18.12)](#equation-eq-second-order) with the first-order matrix difference equation\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix} y_{t+1} \\cr y_{t} \\end{bmatrix}\n",
+ "= \\begin{bmatrix} 2 & - ( 1 - \\sigma) \\cr 1 & 0 \\end{bmatrix} \\begin{bmatrix} y_{t} \\cr y_{t-1} \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "x_{t+1} = M x_t\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "$$\n",
+ "M = \\begin{bmatrix} 2 & - (1 - \\sigma ) \\cr 1 & 0 \\end{bmatrix}, \\quad x_t= \\begin{bmatrix} y_{t} \\cr y_{t-1} \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "Construct an eigendecomposition of $ M $:\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "M = V \\begin{bmatrix} \\lambda_1 & 0 \\cr 0 & \\lambda_2 \\end{bmatrix} V^{-1} \\tag{18.17}\n",
+ "$$\n",
+ "\n",
+ "where columns of $ V $ are eigenvectors corresponding to eigenvalues $ \\lambda_1 $ and $ \\lambda_2 $.\n",
+ "\n",
+ "The eigenvalues can be ordered so that $ \\lambda_1 > 1 > \\lambda_2 $.\n",
+ "\n",
+ "Write equation [(18.12)](#equation-eq-second-order) as\n",
+ "\n",
+ "$$\n",
+ "x_{t+1} = V \\Lambda V^{-1} x_t\n",
+ "$$\n",
+ "\n",
+ "Now we implement the algorithm above.\n",
+ "\n",
+ "First we write a function that iterates $ M $"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "aa1db809",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def iterate_M(x_0, M, num_steps, dtype=np.float64):\n",
+ " \n",
+ " # Eigendecomposition of M\n",
+ " Λ, V = np.linalg.eig(M)\n",
+ " V_inv = np.linalg.inv(V)\n",
+ " \n",
+ " # Initialize the array to store results\n",
+ " xs = np.zeros((x_0.shape[0], \n",
+ " num_steps + 1))\n",
+ " \n",
+ " # Perform the iterations\n",
+ " xs[:, 0] = x_0\n",
+ " for t in range(num_steps):\n",
+ " xs[:, t + 1] = M @ xs[:, t]\n",
+ " \n",
+ " return xs, Λ, V, V_inv\n",
+ "\n",
+ "# Define the state transition matrix M\n",
+ "M = np.array([\n",
+ " [2, -(1 - σ)],\n",
+ " [1, 0]])\n",
+ "\n",
+ "# Initial condition vector x_0\n",
+ "x_0 = np.array([2, 2])\n",
+ "\n",
+ "# Perform the iteration\n",
+ "xs, Λ, V, V_inv = iterate_M(x_0, M, num_steps=100)\n",
+ "\n",
+ "print(f\"eigenvalues:\\n{Λ}\")\n",
+ "print(f\"eigenvectors:\\n{V}\")\n",
+ "print(f\"inverse eigenvectors:\\n{V_inv}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "405ece43",
+ "metadata": {},
+ "source": [
+ "Let’s compare the eigenvalues to the roots [(18.15)](#equation-eq-secretweapon) of equation\n",
+ "[(18.13)](#equation-eq-cha-eq0) that we computed above."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f7588436",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "roots = solve_λs((1, -2, (1 - σ)))\n",
+ "print(f\"roots: {np.round(roots, 8)}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "be94bb93",
+ "metadata": {},
+ "source": [
+ "Hence we confirmed [(18.17)](#equation-eq-eigen-sqrt).\n",
+ "\n",
+ "Information about the square root we are after is also contained\n",
+ "in the two eigenvectors.\n",
+ "\n",
+ "Indeed, each eigenvector is just a two-dimensional subspace of $ {\\mathbb R}^3 $ pinned down by dynamics of the form\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_{t} = \\lambda_i y_{t-1}, \\quad i = 1, 2 \\tag{18.18}\n",
+ "$$\n",
+ "\n",
+ "that we encountered above in equation [(18.8)](#equation-eq-2diff8) above.\n",
+ "\n",
+ "In equation [(18.18)](#equation-eq-invariantsub101), the $ i $th $ \\lambda_i $ equals the $ V_{i, 1}/V_{i,2} $.\n",
+ "\n",
+ "The following graph verifies this for our example."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8e63d31d",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Plotting the eigenvectors\n",
+ "plt.figure(figsize=(8, 8))\n",
+ "\n",
+ "plt.quiver(0, 0, V[0, 0], V[1, 0], angles='xy', scale_units='xy', \n",
+ " scale=1, color='C0', label=fr'$\\lambda_1={np.round(Λ[0], 4)}$')\n",
+ "plt.quiver(0, 0, V[0, 1], V[1, 1], angles='xy', scale_units='xy', \n",
+ " scale=1, color='C1', label=fr'$\\lambda_2={np.round(Λ[1], 4)}$')\n",
+ "\n",
+ "# Annotating the slopes\n",
+ "plt.text(V[0, 0]-0.5, V[1, 0]*1.2, \n",
+ " r'slope=$\\frac{V_{1,1}}{V_{1,2}}=$'+f'{np.round(V[0, 0] / V[1, 0], 4)}', \n",
+ " fontsize=12, color='C0')\n",
+ "plt.text(V[0, 1]-0.5, V[1, 1]*1.2, \n",
+ " r'slope=$\\frac{V_{2,1}}{V_{2,2}}=$'+f'{np.round(V[0, 1] / V[1, 1], 4)}', \n",
+ " fontsize=12, color='C1')\n",
+ "\n",
+ "# Adding labels\n",
+ "plt.axhline(0, color='grey', linewidth=0.5, alpha=0.4)\n",
+ "plt.axvline(0, color='grey', linewidth=0.5, alpha=0.4)\n",
+ "plt.legend()\n",
+ "\n",
+ "plt.xlim(-1.5, 1.5)\n",
+ "plt.ylim(-1.5, 1.5)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0739f0ba",
+ "metadata": {},
+ "source": [
+ "## Invariant subspace approach\n",
+ "\n",
+ "The preceding calculation indicates that we can use the eigenvectors $ V $ to construct 2-dimensional *invariant subspaces*.\n",
+ "\n",
+ "We’ll pursue that possibility now.\n",
+ "\n",
+ "Define the transformed variables\n",
+ "\n",
+ "$$\n",
+ "x_t^* = V^{-1} x_t\n",
+ "$$\n",
+ "\n",
+ "Evidently, we can recover $ x_t $ from $ x_t^* $:\n",
+ "\n",
+ "$$\n",
+ "x_t = V x_t^*\n",
+ "$$\n",
+ "\n",
+ "The following notations and equations will help us.\n",
+ "\n",
+ "Let\n",
+ "\n",
+ "$$\n",
+ "V = \\begin{bmatrix} V_{1,1} & V_{1,2} \\cr \n",
+ " V_{2,1} & V_{2,2} \\end{bmatrix}, \\quad\n",
+ "V^{-1} = \\begin{bmatrix} V^{1,1} & V^{1,2} \\cr \n",
+ " V^{2,1} & V^{2,2} \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "Notice that it follows from\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix} V^{1,1} & V^{1,2} \\cr \n",
+ " V^{2,1} & V^{2,2} \\end{bmatrix} \\begin{bmatrix} V_{1,1} & V_{1,2} \\cr \n",
+ " V_{2,1} & V_{2,2} \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\cr 0 & 1 \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "that\n",
+ "\n",
+ "$$\n",
+ "V^{2,1} V_{1,1} + V^{2,2} V_{2,1} = 0\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "$$\n",
+ "V^{1,1}V_{1,2} + V^{1,2} V_{2,2} = 0.\n",
+ "$$\n",
+ "\n",
+ "These equations will be very useful soon.\n",
+ "\n",
+ "Notice that\n",
+ "\n",
+ "$$\n",
+ "\\begin{bmatrix} x_{1,t+1}^* \\cr x_{2,t+1}^* \\end{bmatrix} = \\begin{bmatrix} \\lambda_1 & 0 \\cr 0 & \\lambda_2 \\end{bmatrix}\n",
+ "\\begin{bmatrix} x_{1,t}^* \\cr x_{2,t}^* \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "To deactivate $ \\lambda_1 $ we want to set\n",
+ "\n",
+ "$$\n",
+ "x_{1,0}^* = 0.\n",
+ "$$\n",
+ "\n",
+ "This can be achieved by setting\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "x_{2,0} = -( V^{1,2})^{-1} V^{1,1} x_{1,0} = V_{2,2} V_{1,2}^{-1} x_{1,0}. \\tag{18.19}\n",
+ "$$\n",
+ "\n",
+ "To deactivate $ \\lambda_2 $, we want to set\n",
+ "\n",
+ "$$\n",
+ "x_{2,0}^* = 0\n",
+ "$$\n",
+ "\n",
+ "This can be achieved by setting\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "x_{2,0} = -(V^{2,2})^{-1} V^{2,1} x_{1,0} = V_{2,1} V_{1,1}^{-1} x_{1,0}. \\tag{18.20}\n",
+ "$$\n",
+ "\n",
+ "Let’s verify [(18.19)](#equation-eq-deactivate1) and [(18.20)](#equation-eq-deactivate2) below\n",
+ "\n",
+ "To deactivate $ \\lambda_1 $ we use [(18.19)](#equation-eq-deactivate1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7d0968ba",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "xd_1 = np.array((x_0[0], \n",
+ " V[1,1]/V[0,1] * x_0[0]),\n",
+ " dtype=np.float64)\n",
+ "\n",
+ "# Compute x_{1,0}^*\n",
+ "np.round(V_inv @ xd_1, 8)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "61b64664",
+ "metadata": {},
+ "source": [
+ "We find $ x_{1,0}^* = 0 $.\n",
+ "\n",
+ "Now we deactivate $ \\lambda_2 $ using [(18.20)](#equation-eq-deactivate2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "282f14aa",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "xd_2 = np.array((x_0[0], \n",
+ " V[1,0]/V[0,0] * x_0[0]), \n",
+ " dtype=np.float64)\n",
+ "\n",
+ "# Compute x_{2,0}^*\n",
+ "np.round(V_inv @ xd_2, 8)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "edcbaaeb",
+ "metadata": {},
+ "source": [
+ "We find $ x_{2,0}^* = 0 $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c7b61009",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Simulate with muted λ1 λ2.\n",
+ "num_steps = 10\n",
+ "xs_λ1 = iterate_M(xd_1, M, num_steps)[0]\n",
+ "xs_λ2 = iterate_M(xd_2, M, num_steps)[0]\n",
+ "\n",
+ "# Compute ratios y_t / y_{t-1}\n",
+ "ratios_λ1 = xs_λ1[1, 1:] / xs_λ1[1, :-1]\n",
+ "ratios_λ2 = xs_λ2[1, 1:] / xs_λ2[1, :-1] "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8c73b2de",
+ "metadata": {},
+ "source": [
+ "The following graph shows the ratios $ y_t / y_{t-1} $ for the two cases.\n",
+ "\n",
+ "We find that the ratios converge to $ \\lambda_2 $ in the first case and $ \\lambda_1 $ in the second case."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "56c1c095",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Plot the ratios for y_t / y_{t-1}\n",
+ "fig, axs = plt.subplots(1, 2, figsize=(12, 6), dpi=500)\n",
+ "\n",
+ "# First subplot\n",
+ "axs[0].plot(np.round(ratios_λ1, 6), \n",
+ " label=r'$\\frac{y_t}{y_{t-1}}$', linewidth=3)\n",
+ "axs[0].axhline(y=Λ[1], color='red', linestyle='--', \n",
+ " label=r'$\\lambda_2$', alpha=0.5)\n",
+ "axs[0].set_xlabel('t', size=18)\n",
+ "axs[0].set_ylabel(r'$\\frac{y_t}{y_{t-1}}$', size=18)\n",
+ "axs[0].set_title(r'$\\frac{y_t}{y_{t-1}}$ after Muting $\\lambda_1$', \n",
+ " size=13)\n",
+ "axs[0].legend()\n",
+ "\n",
+ "# Second subplot\n",
+ "axs[1].plot(ratios_λ2, label=r'$\\frac{y_t}{y_{t-1}}$', \n",
+ " linewidth=3)\n",
+ "axs[1].axhline(y=Λ[0], color='green', linestyle='--', \n",
+ " label=r'$\\lambda_1$', alpha=0.5)\n",
+ "axs[1].set_xlabel('t', size=18)\n",
+ "axs[1].set_ylabel(r'$\\frac{y_t}{y_{t-1}}$', size=18)\n",
+ "axs[1].set_title(r'$\\frac{y_t}{y_{t-1}}$ after Muting $\\lambda_2$', \n",
+ " size=13)\n",
+ "axs[1].legend()\n",
+ "\n",
+ "plt.tight_layout()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cac9e5c6",
+ "metadata": {},
+ "source": [
+ "## Concluding remarks\n",
+ "\n",
+ "This lecture sets the stage for many other applications of the *invariant subspace* methods.\n",
+ "\n",
+ "All of these exploit very similar equations based on eigen decompositions.\n",
+ "\n",
+ "We shall encounter equations very similar to [(18.19)](#equation-eq-deactivate1) and [(18.20)](#equation-eq-deactivate2)\n",
+ "in [Money Financed Government Deficits and Price Levels](https://intro.quantecon.org/money_inflation.html) and in many other places in dynamic economic theory."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9c04c621",
+ "metadata": {},
+ "source": [
+ "## Exercise"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d3eb6eba",
+ "metadata": {},
+ "source": [
+ "## Exercise 18.1\n",
+ "\n",
+ "Please use matrix algebra to formulate the method described by Bertrand Russell at the beginning of this lecture.\n",
+ "\n",
+ "1. Define a state vector $ x_t = \\begin{bmatrix} a_t \\cr b_t \\end{bmatrix} $. \n",
+ "1. Formulate a first-order vector difference equation for $ x_t $ of the form $ x_{t+1} = A x_t $ and\n",
+ " compute the matrix $ A $. \n",
+ "1. Use the system $ x_{t+1} = A x_t $ to replicate the sequence of $ a_t $’s and $ b_t $’s described by Bertrand Russell. \n",
+ "1. Compute the eigenvectors and eigenvalues of $ A $ and compare them to corresponding objects computed in the text of this lecture. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "87a342ce",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 18.1](https://intro.quantecon.org/#greek_square_ex_a)\n",
+ "\n",
+ "Here is one soluition.\n",
+ "\n",
+ "According to the quote, we can formulate\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "a_{t+1} &= a_t + b_t \\\\\n",
+ "b_{t+1} &= 2a_t + b_t\n",
+ "\\end{aligned} \\tag{18.21}\n",
+ "$$\n",
+ "\n",
+ "with $ x_0 = \\begin{bmatrix} a_0 \\cr b_0 \\end{bmatrix} = \\begin{bmatrix} 1 \\cr 1 \\end{bmatrix} $\n",
+ "\n",
+ "By [(18.21)](#equation-eq-gs-ex1system), we can write matrix $ A $ as\n",
+ "\n",
+ "$$\n",
+ "A = \\begin{bmatrix} 1 & 1 \\cr \n",
+ " 2 & 1 \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "Then $ x_{t+1} = A x_t $ for $ t \\in \\{0, \\dots, 5\\} $"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "913b280f",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Define the matrix A\n",
+ "A = np.array([[1, 1],\n",
+ " [2, 1]])\n",
+ "\n",
+ "# Initial vector x_0\n",
+ "x_0 = np.array([1, 1])\n",
+ "\n",
+ "# Number of iterations\n",
+ "n = 6\n",
+ "\n",
+ "# Generate the sequence\n",
+ "xs = np.array([x_0])\n",
+ "x_t = x_0\n",
+ "for _ in range(1, n):\n",
+ " x_t = A @ x_t\n",
+ " xs = np.vstack([xs, x_t])\n",
+ "\n",
+ "# Print the sequence\n",
+ "for i, (a_t, b_t) in enumerate(xs):\n",
+ " print(f\"Iter {i}: a_t = {a_t}, b_t = {b_t}\")\n",
+ "\n",
+ "# Compute eigenvalues and eigenvectors of A\n",
+ "eigenvalues, eigenvectors = np.linalg.eig(A)\n",
+ "\n",
+ "print(f'\\nEigenvalues:\\n{eigenvalues}')\n",
+ "print(f'\\nEigenvectors:\\n{eigenvectors}')"
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476280.9146044,
+ "filename": "greek_square.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Computing Square Roots"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/heavy_tails.ipynb b/_notebooks/heavy_tails.ipynb
new file mode 100644
index 000000000..8c97abc26
--- /dev/null
+++ b/_notebooks/heavy_tails.ipynb
@@ -0,0 +1,1939 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "2173274c",
+ "metadata": {},
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c96ec8e5",
+ "metadata": {},
+ "source": [
+ "# Heavy-Tailed Distributions\n",
+ "\n",
+ "In addition to what’s in Anaconda, this lecture will need the following libraries:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "cbe3dd25",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "!pip install --upgrade yfinance wbgapi"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3f7db1ec",
+ "metadata": {},
+ "source": [
+ "We use the following imports."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7dff45a0",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np\n",
+ "import yfinance as yf\n",
+ "import pandas as pd\n",
+ "import statsmodels.api as sm\n",
+ "\n",
+ "import wbgapi as wb\n",
+ "from scipy.stats import norm, cauchy\n",
+ "from pandas.plotting import register_matplotlib_converters\n",
+ "register_matplotlib_converters()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f499b33f",
+ "metadata": {},
+ "source": [
+ "## Overview\n",
+ "\n",
+ "Heavy-tailed distributions are a class of distributions that generate “extreme” outcomes.\n",
+ "\n",
+ "In the natural sciences (and in more traditional economics courses), heavy-tailed distributions are seen as quite exotic and non-standard.\n",
+ "\n",
+ "However, it turns out that heavy-tailed distributions play a crucial role in economics.\n",
+ "\n",
+ "In fact many – if not most – of the important distributions in economics are heavy-tailed.\n",
+ "\n",
+ "In this lecture we explain what heavy tails are and why they are – or at least\n",
+ "why they should be – central to economic analysis."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fb237e07",
+ "metadata": {},
+ "source": [
+ "### Introduction: light tails\n",
+ "\n",
+ "Most [commonly used probability distributions](https://intro.quantecon.org/prob_dist.html) in classical statistics and\n",
+ "the natural sciences have “light tails.”\n",
+ "\n",
+ "To explain this concept, let’s look first at examples."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "17faf690",
+ "metadata": {},
+ "source": [
+ "### \n",
+ "\n",
+ "The classic example is the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution), which has density\n",
+ "\n",
+ "$$\n",
+ "f(x) = \\frac{1}{\\sqrt{2\\pi}\\sigma} \n",
+ "\\exp\\left( -\\frac{(x-\\mu)^2}{2 \\sigma^2} \\right)\n",
+ "\\qquad\n",
+ "(-\\infty < x < \\infty)\n",
+ "$$\n",
+ "\n",
+ "The two parameters $ \\mu $ and $ \\sigma $ are the mean and standard deviation\n",
+ "respectively.\n",
+ "\n",
+ "As $ x $ deviates from $ \\mu $, the value of $ f(x) $ goes to zero extremely\n",
+ "quickly.\n",
+ "\n",
+ "We can see this when we plot the density and show a histogram of observations,\n",
+ "as with the following code (which assumes $ \\mu=0 $ and $ \\sigma=1 $)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "03e27a5d",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "X = norm.rvs(size=1_000_000)\n",
+ "ax.hist(X, bins=40, alpha=0.4, label='histogram', density=True)\n",
+ "x_grid = np.linspace(-4, 4, 400)\n",
+ "ax.plot(x_grid, norm.pdf(x_grid), label='density')\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e651dbb4",
+ "metadata": {},
+ "source": [
+ "Notice how\n",
+ "\n",
+ "- the density’s tails converge quickly to zero in both directions and \n",
+ "- even with 1,000,000 draws, we get no very large or very small observations. \n",
+ "\n",
+ "\n",
+ "We can see the last point more clearly by executing"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "2668bed4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "X.min(), X.max()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7efafb1a",
+ "metadata": {},
+ "source": [
+ "Here’s another view of draws from the same distribution:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "2734506c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "n = 2000\n",
+ "fig, ax = plt.subplots()\n",
+ "data = norm.rvs(size=n)\n",
+ "ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ "ax.vlines(list(range(n)), 0, data, lw=0.2)\n",
+ "ax.set_ylim(-15, 15)\n",
+ "ax.set_xlabel('$i$')\n",
+ "ax.set_ylabel('$X_i$', rotation=0)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4bd54992",
+ "metadata": {},
+ "source": [
+ "We have plotted each individual draw $ X_i $ against $ i $.\n",
+ "\n",
+ "None are very large or very small.\n",
+ "\n",
+ "In other words, extreme observations are rare and draws tend not to deviate\n",
+ "too much from the mean.\n",
+ "\n",
+ "Putting this another way, light-tailed distributions are those that\n",
+ "rarely generate extreme values.\n",
+ "\n",
+ "(A more formal definition is given [below](#heavy-tail-formal-definition).)\n",
+ "\n",
+ "Many statisticians and econometricians\n",
+ "use rules of thumb such as “outcomes more than four or five\n",
+ "standard deviations from the mean can safely be ignored.”\n",
+ "\n",
+ "But this is only true when distributions have light tails."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4f059a9a",
+ "metadata": {},
+ "source": [
+ "### When are light tails valid?\n",
+ "\n",
+ "In probability theory and in the real world, many distributions are\n",
+ "light-tailed.\n",
+ "\n",
+ "For example, human height is light-tailed.\n",
+ "\n",
+ "Yes, it’s true that we see some very tall people.\n",
+ "\n",
+ "- For example, basketballer [Sun Mingming](https://en.wikipedia.org/wiki/Sun_Mingming) is 2.32 meters tall \n",
+ "\n",
+ "\n",
+ "But have you ever heard of someone who is 20 meters tall? Or 200? Or 2000?\n",
+ "\n",
+ "Have you ever wondered why not?\n",
+ "\n",
+ "After all, there are 8 billion people in the world!\n",
+ "\n",
+ "In essence, the reason we don’t see such draws is that the distribution of\n",
+ "human height has very light tails.\n",
+ "\n",
+ "In fact the distribution of human height obeys a bell-shaped curve similar to the normal distribution."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "114c1e0f",
+ "metadata": {},
+ "source": [
+ "### Returns on assets\n",
+ "\n",
+ "But what about economic data?\n",
+ "\n",
+ "Let’s look at some financial data first.\n",
+ "\n",
+ "Our aim is to plot the daily change in the price of Amazon (AMZN) stock for\n",
+ "the period from 1st January 2015 to 1st July 2022.\n",
+ "\n",
+ "This equates to daily returns if we set dividends aside.\n",
+ "\n",
+ "The code below produces the desired plot using Yahoo financial data via the `yfinance` library."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3e015f2b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data = yf.download('AMZN', '2015-1-1', '2022-7-1')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "03f1c308",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "s = data['Close']\n",
+ "r = s.pct_change()\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "\n",
+ "ax.plot(r, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ "ax.vlines(r.index, 0, r.values, lw=0.2)\n",
+ "ax.set_ylabel('returns', fontsize=12)\n",
+ "ax.set_xlabel('date', fontsize=12)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6f57b8a7",
+ "metadata": {},
+ "source": [
+ "This data looks different to the draws from the normal distribution we saw above.\n",
+ "\n",
+ "Several of observations are quite extreme.\n",
+ "\n",
+ "We get a similar picture if we look at other assets, such as Bitcoin"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1f561eb7",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data = yf.download('BTC-USD', '2015-1-1', '2022-7-1')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ce165801",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "s = data['Close']\n",
+ "r = s.pct_change()\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "\n",
+ "ax.plot(r, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ "ax.vlines(r.index, 0, r.values, lw=0.2)\n",
+ "ax.set_ylabel('returns', fontsize=12)\n",
+ "ax.set_xlabel('date', fontsize=12)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "95ddb0d3",
+ "metadata": {},
+ "source": [
+ "The histogram also looks different to the histogram of the normal\n",
+ "distribution:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3ea1b8d2",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "r = np.random.standard_t(df=5, size=1000)\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.hist(r, bins=60, alpha=0.4, label='bitcoin returns', density=True)\n",
+ "\n",
+ "xmin, xmax = plt.xlim()\n",
+ "x = np.linspace(xmin, xmax, 100)\n",
+ "p = norm.pdf(x, np.mean(r), np.std(r))\n",
+ "ax.plot(x, p, linewidth=2, label='normal distribution')\n",
+ "\n",
+ "ax.set_xlabel('returns', fontsize=12)\n",
+ "ax.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e256e613",
+ "metadata": {},
+ "source": [
+ "If we look at higher frequency returns data (e.g., tick-by-tick), we often see\n",
+ "even more extreme observations.\n",
+ "\n",
+ "See, for example, [[Mandelbrot, 1963](https://intro.quantecon.org/zreferences.html#id86)] or [[Rachev, 2003](https://intro.quantecon.org/zreferences.html#id85)]."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "80bb98d9",
+ "metadata": {},
+ "source": [
+ "### Other data\n",
+ "\n",
+ "The data we have just seen is said to be “heavy-tailed”.\n",
+ "\n",
+ "With heavy-tailed distributions, extreme outcomes occur relatively\n",
+ "frequently."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dcd34e98",
+ "metadata": {},
+ "source": [
+ "### \n",
+ "\n",
+ "Importantly, there are many examples of heavy-tailed distributions\n",
+ "observed in economic and financial settings!\n",
+ "\n",
+ "For example, the income and the wealth distributions are heavy-tailed\n",
+ "\n",
+ "- You can imagine this: most people have low or modest wealth but some people\n",
+ " are extremely rich. \n",
+ "\n",
+ "\n",
+ "The firm size distribution is also heavy-tailed\n",
+ "\n",
+ "- You can imagine this too: most firms are small but some firms are enormous. \n",
+ "\n",
+ "\n",
+ "The distribution of town and city sizes is heavy-tailed\n",
+ "\n",
+ "- Most towns and cities are small but some are very large. \n",
+ "\n",
+ "\n",
+ "Later in this lecture, we examine heavy tails in these distributions."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7c9110f3",
+ "metadata": {},
+ "source": [
+ "### Why should we care?\n",
+ "\n",
+ "Heavy tails are common in economic data but does that mean they are important?\n",
+ "\n",
+ "The answer to this question is affirmative!\n",
+ "\n",
+ "When distributions are heavy-tailed, we need to think carefully about issues\n",
+ "like\n",
+ "\n",
+ "- diversification and risk \n",
+ "- forecasting \n",
+ "- taxation (across a heavy-tailed income distribution), etc. \n",
+ "\n",
+ "\n",
+ "We return to these points [below](#heavy-tail-application)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b6994896",
+ "metadata": {},
+ "source": [
+ "## Visual comparisons\n",
+ "\n",
+ "In this section, we will introduce important concepts such as the Pareto distribution, Counter CDFs, and Power laws, which aid in recognizing heavy-tailed distributions.\n",
+ "\n",
+ "Later we will provide a mathematical definition of the difference between\n",
+ "light and heavy tails.\n",
+ "\n",
+ "But for now let’s do some visual comparisons to help us build intuition on the\n",
+ "difference between these two types of distributions."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "80d01ff0",
+ "metadata": {},
+ "source": [
+ "### Simulations\n",
+ "\n",
+ "The figure below shows a simulation.\n",
+ "\n",
+ "The top two subfigures each show 120 independent draws from the normal\n",
+ "distribution, which is light-tailed.\n",
+ "\n",
+ "The bottom subfigure shows 120 independent draws from [the Cauchy\n",
+ "distribution](https://en.wikipedia.org/wiki/Cauchy_distribution), which is\n",
+ "heavy-tailed."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7fa810d6",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "n = 120\n",
+ "np.random.seed(11)\n",
+ "\n",
+ "fig, axes = plt.subplots(3, 1, figsize=(6, 12))\n",
+ "\n",
+ "for ax in axes:\n",
+ " ax.set_ylim((-120, 120))\n",
+ "\n",
+ "s_vals = 2, 12\n",
+ "\n",
+ "for ax, s in zip(axes[:2], s_vals):\n",
+ " data = np.random.randn(n) * s\n",
+ " ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ " ax.vlines(list(range(n)), 0, data, lw=0.2)\n",
+ " ax.set_title(fr\"draws from $N(0, \\sigma^2)$ with $\\sigma = {s}$\", fontsize=11)\n",
+ "\n",
+ "ax = axes[2]\n",
+ "distribution = cauchy()\n",
+ "data = distribution.rvs(n)\n",
+ "ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ "ax.vlines(list(range(n)), 0, data, lw=0.2)\n",
+ "ax.set_title(f\"draws from the Cauchy distribution\", fontsize=11)\n",
+ "\n",
+ "plt.subplots_adjust(hspace=0.25)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0bc351cf",
+ "metadata": {},
+ "source": [
+ "In the top subfigure, the standard deviation of the normal distribution is 2,\n",
+ "and the draws are clustered around the mean.\n",
+ "\n",
+ "In the middle subfigure, the standard deviation is increased to 12 and, as\n",
+ "expected, the amount of dispersion rises.\n",
+ "\n",
+ "The bottom subfigure, with the Cauchy draws, shows a different pattern: tight\n",
+ "clustering around the mean for the great majority of observations, combined\n",
+ "with a few sudden large deviations from the mean.\n",
+ "\n",
+ "This is typical of a heavy-tailed distribution."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9651db67",
+ "metadata": {},
+ "source": [
+ "### Nonnegative distributions\n",
+ "\n",
+ "Let’s compare some distributions that only take nonnegative values.\n",
+ "\n",
+ "One is the exponential distribution, which we discussed in [our lecture on probability and distributions](https://intro.quantecon.org/prob_dist.html).\n",
+ "\n",
+ "The exponential distribution is a light-tailed distribution.\n",
+ "\n",
+ "Here are some draws from the exponential distribution."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "768f74b8",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "n = 120\n",
+ "np.random.seed(11)\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.set_ylim((0, 50))\n",
+ "\n",
+ "data = np.random.exponential(size=n)\n",
+ "ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ "ax.vlines(list(range(n)), 0, data, lw=0.2)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b32e2557",
+ "metadata": {},
+ "source": [
+ "Another nonnegative distribution is the [Pareto distribution](https://en.wikipedia.org/wiki/Pareto_distribution).\n",
+ "\n",
+ "If $ X $ has the Pareto distribution, then there are positive constants $ \\bar x $\n",
+ "and $ \\alpha $ such that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\mathbb P\\{X > x\\} =\n",
+ "\\begin{cases}\n",
+ " \\left( \\bar x/x \\right)^{\\alpha}\n",
+ " & \\text{ if } x \\geq \\bar x\n",
+ " \\\\\n",
+ " 1\n",
+ " & \\text{ if } x < \\bar x\n",
+ "\\end{cases} \\tag{22.1}\n",
+ "$$\n",
+ "\n",
+ "The parameter $ \\alpha $ is called the **tail index** and $ \\bar x $ is called the\n",
+ "**minimum**.\n",
+ "\n",
+ "The Pareto distribution is a heavy-tailed distribution.\n",
+ "\n",
+ "One way that the Pareto distribution arises is as the exponential of an\n",
+ "exponential random variable.\n",
+ "\n",
+ "In particular, if $ X $ is exponentially distributed with rate parameter $ \\alpha $, then\n",
+ "\n",
+ "$$\n",
+ "Y = \\bar x \\exp(X)\n",
+ "$$\n",
+ "\n",
+ "is Pareto-distributed with minimum $ \\bar x $ and tail index $ \\alpha $.\n",
+ "\n",
+ "Here are some draws from the Pareto distribution with tail index $ 1 $ and minimum\n",
+ "$ 1 $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "18289b60",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "n = 120\n",
+ "np.random.seed(11)\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.set_ylim((0, 80))\n",
+ "exponential_data = np.random.exponential(size=n)\n",
+ "pareto_data = np.exp(exponential_data)\n",
+ "ax.plot(list(range(n)), pareto_data, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ "ax.vlines(list(range(n)), 0, pareto_data, lw=0.2)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4169eceb",
+ "metadata": {},
+ "source": [
+ "Notice how extreme outcomes are more common."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1ae5ff1f",
+ "metadata": {},
+ "source": [
+ "### Counter CDFs\n",
+ "\n",
+ "For nonnegative random variables, one way to visualize the difference between\n",
+ "light and heavy tails is to look at the\n",
+ "**counter CDF** (CCDF).\n",
+ "\n",
+ "For a random variable $ X $ with CDF $ F $, the CCDF is the function\n",
+ "\n",
+ "$$\n",
+ "G(x) := 1 - F(x) = \\mathbb P\\{X > x\\}\n",
+ "$$\n",
+ "\n",
+ "(Some authors call $ G $ the “survival” function.)\n",
+ "\n",
+ "The CCDF shows how fast the upper tail goes to zero as $ x \\to \\infty $.\n",
+ "\n",
+ "If $ X $ is exponentially distributed with rate parameter $ \\alpha $, then the CCDF is\n",
+ "\n",
+ "$$\n",
+ "G_E(x) = \\exp(- \\alpha x)\n",
+ "$$\n",
+ "\n",
+ "This function goes to zero relatively quickly as $ x $ gets large.\n",
+ "\n",
+ "The standard Pareto distribution, where $ \\bar x = 1 $, has CCDF\n",
+ "\n",
+ "$$\n",
+ "G_P(x) = x^{- \\alpha}\n",
+ "$$\n",
+ "\n",
+ "This function goes to zero as $ x \\to \\infty $, but much slower than $ G_E $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "283e3ee7",
+ "metadata": {},
+ "source": [
+ "### Exercise 22.1\n",
+ "\n",
+ "Show how the CCDF of the standard Pareto distribution can be derived from the CCDF of the exponential distribution."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e08f8128",
+ "metadata": {},
+ "source": [
+ "### Solution to[ Exercise 22.1](https://intro.quantecon.org/#ht_ex_x1)\n",
+ "\n",
+ "Letting $ G_E $ and $ G_P $ be defined as above, letting $ X $ be exponentially\n",
+ "distributed with rate parameter $ \\alpha $, and letting $ Y = \\exp(X) $, we have\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ " G_P(y) & = \\mathbb P\\{Y > y\\} \\\\\n",
+ " & = \\mathbb P\\{\\exp(X) > y\\} \\\\\n",
+ " & = \\mathbb P\\{X > \\ln y\\} \\\\\n",
+ " & = G_E(\\ln y) \\\\\n",
+ " & = \\exp( - \\alpha \\ln y) \\\\\n",
+ " & = y^{-\\alpha}\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "Here’s a plot that illustrates how $ G_E $ goes to zero faster than $ G_P $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9a9cd13b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "x = np.linspace(1.5, 100, 1000)\n",
+ "fig, ax = plt.subplots()\n",
+ "alpha = 1.0\n",
+ "ax.plot(x, np.exp(- alpha * x), label='exponential', alpha=0.8)\n",
+ "ax.plot(x, x**(- alpha), label='Pareto', alpha=0.8)\n",
+ "ax.set_xlabel('X value')\n",
+ "ax.set_ylabel('CCDF')\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "327ab90a",
+ "metadata": {},
+ "source": [
+ "Here’s a log-log plot of the same functions, which makes visual comparison\n",
+ "easier."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "54ca57aa",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "alpha = 1.0\n",
+ "ax.loglog(x, np.exp(- alpha * x), label='exponential', alpha=0.8)\n",
+ "ax.loglog(x, x**(- alpha), label='Pareto', alpha=0.8)\n",
+ "ax.set_xlabel('log value')\n",
+ "ax.set_ylabel('log prob')\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "507a3d98",
+ "metadata": {},
+ "source": [
+ "In the log-log plot, the Pareto CCDF is linear, while the exponential one is\n",
+ "concave.\n",
+ "\n",
+ "This idea is often used to separate light- and heavy-tailed distributions in\n",
+ "visualisations — we return to this point below."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fb3ffc3a",
+ "metadata": {},
+ "source": [
+ "### Empirical CCDFs\n",
+ "\n",
+ "The sample counterpart of the CCDF function is the **empirical CCDF**.\n",
+ "\n",
+ "Given a sample $ x_1, \\ldots, x_n $, the empirical CCDF is given by\n",
+ "\n",
+ "$$\n",
+ "\\hat G(x) = \\frac{1}{n} \\sum_{i=1}^n \\mathbb 1\\{x_i > x\\}\n",
+ "$$\n",
+ "\n",
+ "Thus, $ \\hat G(x) $ shows the fraction of the sample that exceeds $ x $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "91dc7022",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def eccdf(x, data):\n",
+ " \"Simple empirical CCDF function.\"\n",
+ " return np.mean(data > x)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d108cf2b",
+ "metadata": {},
+ "source": [
+ "Here’s a figure containing some empirical CCDFs from simulated data."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e4e03f92",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Parameters and grid\n",
+ "x_grid = np.linspace(1, 1000, 1000)\n",
+ "sample_size = 1000\n",
+ "np.random.seed(13)\n",
+ "z = np.random.randn(sample_size)\n",
+ "\n",
+ "# Draws\n",
+ "data_exp = np.random.exponential(size=sample_size)\n",
+ "data_logn = np.exp(z)\n",
+ "data_pareto = np.exp(np.random.exponential(size=sample_size))\n",
+ "\n",
+ "data_list = [data_exp, data_logn, data_pareto]\n",
+ "\n",
+ "# Build figure\n",
+ "fig, axes = plt.subplots(3, 1, figsize=(6, 8))\n",
+ "axes = axes.flatten()\n",
+ "labels = ['exponential', 'lognormal', 'Pareto']\n",
+ "\n",
+ "for data, label, ax in zip(data_list, labels, axes):\n",
+ "\n",
+ " ax.loglog(x_grid, [eccdf(x, data) for x in x_grid], \n",
+ " 'o', markersize=3.0, alpha=0.5, label=label)\n",
+ " ax.set_xlabel(\"log value\")\n",
+ " ax.set_ylabel(\"log prob\")\n",
+ " \n",
+ " ax.legend()\n",
+ " \n",
+ " \n",
+ "fig.subplots_adjust(hspace=0.4)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c9c0e974",
+ "metadata": {},
+ "source": [
+ "As with the CCDF, the empirical CCDF from the Pareto distributions is\n",
+ "approximately linear in a log-log plot.\n",
+ "\n",
+ "We will use this idea [below](https://intro.quantecon.org/heavy_tails.html#heavy-tails-in-economic-cross-sections) when we look at real data."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4735cd8d",
+ "metadata": {},
+ "source": [
+ "#### Q-Q Plots\n",
+ "\n",
+ "We can also use a [qq plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot) to do a visual comparison between two probability distributions.\n",
+ "\n",
+ "The [statsmodels](https://www.statsmodels.org/stable/index.html) package provides a convenient [qqplot](https://www.statsmodels.org/stable/generated/statsmodels.graphics.gofplots.qqplot.html) function that, by default, compares sample data to the quintiles of the normal distribution.\n",
+ "\n",
+ "If the data is drawn from a normal distribution, the plot would look like:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "205dc55a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data_normal = np.random.normal(size=sample_size)\n",
+ "sm.qqplot(data_normal, line='45')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "77c1d2d1",
+ "metadata": {},
+ "source": [
+ "We can now compare this with the exponential, log-normal, and Pareto distributions"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b44a0236",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Build figure\n",
+ "fig, axes = plt.subplots(1, 3, figsize=(12, 4))\n",
+ "axes = axes.flatten()\n",
+ "labels = ['exponential', 'lognormal', 'Pareto']\n",
+ "for data, label, ax in zip(data_list, labels, axes):\n",
+ " sm.qqplot(data, line='45', ax=ax, )\n",
+ " ax.set_title(label)\n",
+ "plt.tight_layout()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f6b54a43",
+ "metadata": {},
+ "source": [
+ "### Power laws\n",
+ "\n",
+ "One specific class of heavy-tailed distributions has been found repeatedly in\n",
+ "economic and social phenomena: the class of so-called power laws.\n",
+ "\n",
+ "A random variable $ X $ is said to have a **power law** if, for some $ \\alpha > 0 $,\n",
+ "\n",
+ "$$\n",
+ "\\mathbb P\\{X > x\\} \\approx x^{-\\alpha}\n",
+ "\\quad \\text{when \\$x\\$ is large}\n",
+ "$$\n",
+ "\n",
+ "We can write this more mathematically as\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\lim_{x \\to \\infty} x^\\alpha \\, \\mathbb P\\{X > x\\} = c\n",
+ "\\quad \\text{for some \\$c > 0\\$} \\tag{22.2}\n",
+ "$$\n",
+ "\n",
+ "It is also common to say that a random variable $ X $ with this property\n",
+ "has a **Pareto tail** with **tail index** $ \\alpha $.\n",
+ "\n",
+ "Notice that every Pareto distribution with tail index $ \\alpha $\n",
+ "has a **Pareto tail** with **tail index** $ \\alpha $.\n",
+ "\n",
+ "We can think of power laws as a generalization of Pareto distributions.\n",
+ "\n",
+ "They are distributions that resemble Pareto distributions in their upper right\n",
+ "tail.\n",
+ "\n",
+ "Another way to think of power laws is a set of distributions with a specific\n",
+ "kind of (very) heavy tail."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c05bb1fb",
+ "metadata": {},
+ "source": [
+ "## Heavy tails in economic cross-sections\n",
+ "\n",
+ "As mentioned above, heavy tails are pervasive in economic data.\n",
+ "\n",
+ "In fact power laws seem to be very common as well.\n",
+ "\n",
+ "We now illustrate this by showing the empirical CCDF of heavy tails.\n",
+ "\n",
+ "All plots are in log-log, so that a power law shows up as a linear log-log\n",
+ "plot, at least in the upper tail.\n",
+ "\n",
+ "We hide the code that generates the figures, which is somewhat complex, but\n",
+ "readers are of course welcome to explore the code (perhaps after examining the figures)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f1f38829",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def empirical_ccdf(data, \n",
+ " ax, \n",
+ " aw=None, # weights\n",
+ " label=None,\n",
+ " xlabel=None,\n",
+ " add_reg_line=False, \n",
+ " title=None):\n",
+ " \"\"\"\n",
+ " Take data vector and return prob values for plotting.\n",
+ " Upgraded empirical_ccdf\n",
+ " \"\"\"\n",
+ " y_vals = np.empty_like(data, dtype='float64')\n",
+ " p_vals = np.empty_like(data, dtype='float64')\n",
+ " n = len(data)\n",
+ " if aw is None:\n",
+ " for i, d in enumerate(data):\n",
+ " # record fraction of sample above d\n",
+ " y_vals[i] = np.sum(data >= d) / n\n",
+ " p_vals[i] = np.sum(data == d) / n\n",
+ " else:\n",
+ " fw = np.empty_like(aw, dtype='float64')\n",
+ " for i, a in enumerate(aw):\n",
+ " fw[i] = a / np.sum(aw)\n",
+ " pdf = lambda x: np.interp(x, data, fw)\n",
+ " data = np.sort(data)\n",
+ " j = 0\n",
+ " for i, d in enumerate(data):\n",
+ " j += pdf(d)\n",
+ " y_vals[i] = 1- j\n",
+ "\n",
+ " x, y = np.log(data), np.log(y_vals)\n",
+ " \n",
+ " results = sm.OLS(y, sm.add_constant(x)).fit()\n",
+ " b, a = results.params\n",
+ " \n",
+ " kwargs = [('alpha', 0.3)]\n",
+ " if label:\n",
+ " kwargs.append(('label', label))\n",
+ " kwargs = dict(kwargs)\n",
+ "\n",
+ " ax.scatter(x, y, **kwargs)\n",
+ " if add_reg_line:\n",
+ " ax.plot(x, x * a + b, 'k-', alpha=0.6, label=f\"slope = ${a: 1.2f}$\")\n",
+ " if not xlabel:\n",
+ " xlabel='log value'\n",
+ " ax.set_xlabel(xlabel, fontsize=12)\n",
+ " ax.set_ylabel(\"log prob\", fontsize=12)\n",
+ " \n",
+ " if label:\n",
+ " ax.legend(loc='lower left', fontsize=12)\n",
+ " \n",
+ " if title:\n",
+ " ax.set_title(title)\n",
+ " \n",
+ " return np.log(data), y_vals, p_vals"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ff2ad28c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def extract_wb(varlist=['NY.GDP.MKTP.CD'], \n",
+ " c='all', \n",
+ " s=1900, \n",
+ " e=2021, \n",
+ " varnames=None):\n",
+ " \n",
+ " df = wb.data.DataFrame(varlist, economy=c, time=range(s, e+1, 1), skipAggs=True)\n",
+ " df.index.name = 'country'\n",
+ " \n",
+ " if varnames is not None:\n",
+ " df.columns = variable_names\n",
+ "\n",
+ " cntry_mapper = pd.DataFrame(wb.economy.info().items)[['id','value']].set_index('id').to_dict()['value']\n",
+ " df.index = df.index.map(lambda x: cntry_mapper[x]) #map iso3c to name values\n",
+ " \n",
+ " return df"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dc830dfa",
+ "metadata": {},
+ "source": [
+ "### Firm size\n",
+ "\n",
+ "Here is a plot of the firm size distribution for the largest 500 firms in 2020 taken from Forbes Global 2000."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "96aababb",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_fs = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-global2000.csv')\n",
+ "df_fs = df_fs[['Country', 'Sales', 'Profits', 'Assets', 'Market Value']]\n",
+ "fig, ax = plt.subplots(figsize=(6.4, 3.5))\n",
+ "\n",
+ "label=\"firm size (market value)\"\n",
+ "top = 500 # set the cutting for top\n",
+ "d = df_fs.sort_values('Market Value', ascending=False)\n",
+ "empirical_ccdf(np.asarray(d['Market Value'])[:top], ax, label=label, add_reg_line=True)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3a493fba",
+ "metadata": {},
+ "source": [
+ "### City size\n",
+ "\n",
+ "Here are plots of the city size distribution for the US and Brazil in 2023 from the World Population Review.\n",
+ "\n",
+ "The size is measured by population."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f6a52747",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# import population data of cities in 2023 United States and 2023 Brazil from world population review\n",
+ "df_cs_us = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_us.csv')\n",
+ "df_cs_br = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_brazil.csv')\n",
+ "\n",
+ "fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))\n",
+ "\n",
+ "empirical_ccdf(np.asarray(df_cs_us[\"pop2023\"]), axes[0], label=\"US\", add_reg_line=True)\n",
+ "empirical_ccdf(np.asarray(df_cs_br['pop2023']), axes[1], label=\"Brazil\", add_reg_line=True)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "35af160c",
+ "metadata": {},
+ "source": [
+ "### Wealth\n",
+ "\n",
+ "Here is a plot of the upper tail (top 500) of the wealth distribution.\n",
+ "\n",
+ "The data is from the Forbes Billionaires list in 2020."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b51f326d",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_w = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-billionaires.csv')\n",
+ "df_w = df_w[['country', 'realTimeWorth', 'realTimeRank']].dropna()\n",
+ "df_w = df_w.astype({'realTimeRank': int})\n",
+ "df_w = df_w.sort_values('realTimeRank', ascending=True).copy()\n",
+ "countries = ['United States', 'Japan', 'India', 'Italy'] \n",
+ "N = len(countries)\n",
+ "\n",
+ "fig, axs = plt.subplots(2, 2, figsize=(8, 6))\n",
+ "axs = axs.flatten()\n",
+ "\n",
+ "for i, c in enumerate(countries):\n",
+ " df_w_c = df_w[df_w['country'] == c].reset_index()\n",
+ " z = np.asarray(df_w_c['realTimeWorth'])\n",
+ " # print('number of the global richest 2000 from '+ c, len(z))\n",
+ " top = 500 # cut-off number: top 500\n",
+ " if len(z) <= top: \n",
+ " z = z[:top]\n",
+ "\n",
+ " empirical_ccdf(z[:top], axs[i], label=c, xlabel='log wealth', add_reg_line=True)\n",
+ " \n",
+ "fig.tight_layout()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3fdd8222",
+ "metadata": {},
+ "source": [
+ "### GDP\n",
+ "\n",
+ "Of course, not all cross-sectional distributions are heavy-tailed.\n",
+ "\n",
+ "Here we show cross-country per capita GDP."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "cb9cbfb7",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# get gdp and gdp per capita for all regions and countries in 2021\n",
+ "\n",
+ "variable_code = ['NY.GDP.MKTP.CD', 'NY.GDP.PCAP.CD']\n",
+ "variable_names = ['GDP', 'GDP per capita']\n",
+ "\n",
+ "df_gdp1 = extract_wb(varlist=variable_code, \n",
+ " c=\"all\", \n",
+ " s=2021, \n",
+ " e=2021, \n",
+ " varnames=variable_names)\n",
+ "df_gdp1.dropna(inplace=True)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8987581b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))\n",
+ "\n",
+ "for name, ax in zip(variable_names, axes):\n",
+ " empirical_ccdf(np.asarray(df_gdp1[name]).astype(\"float64\"), ax, add_reg_line=False, label=name)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4e718dfc",
+ "metadata": {},
+ "source": [
+ "The plot is concave rather than linear, so the distribution has light tails.\n",
+ "\n",
+ "One reason is that this is data on an aggregate variable, which involves some\n",
+ "averaging in its definition.\n",
+ "\n",
+ "Averaging tends to eliminate extreme outcomes."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0a2ad1da",
+ "metadata": {},
+ "source": [
+ "## Failure of the LLN\n",
+ "\n",
+ "One impact of heavy tails is that sample averages can be poor estimators of\n",
+ "the underlying mean of the distribution.\n",
+ "\n",
+ "To understand this point better, recall [our earlier discussion](https://intro.quantecon.org/lln_clt.html)\n",
+ "of the law of large numbers, which considered IID $ X_1, \\ldots, X_n $ with common distribution $ F $\n",
+ "\n",
+ "If $ \\mathbb E |X_i| $ is finite, then\n",
+ "the sample mean $ \\bar X_n := \\frac{1}{n} \\sum_{i=1}^n X_i $ satisfies\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\mathbb P \\left\\{ \\bar X_n \\to \\mu \\text{ as } n \\to \\infty \\right\\} = 1 \\tag{22.3}\n",
+ "$$\n",
+ "\n",
+ "where $ \\mu := \\mathbb E X_i = \\int x F(dx) $ is the common mean of the sample.\n",
+ "\n",
+ "The condition $ \\mathbb E | X_i | = \\int |x| F(dx) < \\infty $ holds\n",
+ "in most cases but can fail if the distribution $ F $ is very heavy-tailed.\n",
+ "\n",
+ "For example, it fails for the Cauchy distribution.\n",
+ "\n",
+ "Let’s have a look at the behavior of the sample mean in this case, and see\n",
+ "whether or not the LLN is still valid."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fb760a44",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "from scipy.stats import cauchy\n",
+ "\n",
+ "np.random.seed(1234)\n",
+ "N = 1_000\n",
+ "\n",
+ "distribution = cauchy()\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "data = distribution.rvs(N)\n",
+ "\n",
+ "# Compute sample mean at each n\n",
+ "sample_mean = np.empty(N)\n",
+ "for n in range(1, N):\n",
+ " sample_mean[n] = np.mean(data[:n])\n",
+ "\n",
+ "# Plot\n",
+ "ax.plot(range(N), sample_mean, alpha=0.6, label='$\\\\bar{X}_n$')\n",
+ "ax.plot(range(N), np.zeros(N), 'k--', lw=0.5)\n",
+ "ax.set_xlabel(r\"$n$\")\n",
+ "ax.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "77205cb1",
+ "metadata": {},
+ "source": [
+ "The sequence shows no sign of converging.\n",
+ "\n",
+ "We return to this point in the exercises.\n",
+ "\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b9da5f97",
+ "metadata": {},
+ "source": [
+ "## Why do heavy tails matter?\n",
+ "\n",
+ "We have now seen that\n",
+ "\n",
+ "1. heavy tails are frequent in economics and \n",
+ "1. the law of large numbers fails when tails are very heavy. \n",
+ "\n",
+ "\n",
+ "But what about in the real world? Do heavy tails matter?\n",
+ "\n",
+ "Let’s briefly discuss why they do."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d8f85723",
+ "metadata": {},
+ "source": [
+ "### Diversification\n",
+ "\n",
+ "One of the most important ideas in investing is using diversification to\n",
+ "reduce risk.\n",
+ "\n",
+ "This is a very old idea — consider, for example, the expression “don’t put all your eggs in one basket”.\n",
+ "\n",
+ "To illustrate, consider an investor with one dollar of wealth and a choice over\n",
+ "$ n $ assets with payoffs $ X_1, \\ldots, X_n $.\n",
+ "\n",
+ "Suppose that returns on distinct assets are\n",
+ "independent and each return has mean $ \\mu $ and variance $ \\sigma^2 $.\n",
+ "\n",
+ "If the investor puts all wealth in one asset, say, then the expected payoff of the\n",
+ "portfolio is $ \\mu $ and the variance is $ \\sigma^2 $.\n",
+ "\n",
+ "If instead the investor puts share $ 1/n $ of her wealth in each asset, then the portfolio payoff is\n",
+ "\n",
+ "$$\n",
+ "Y_n = \\sum_{i=1}^n \\frac{X_i}{n} = \\frac{1}{n} \\sum_{i=1}^n X_i.\n",
+ "$$\n",
+ "\n",
+ "Try computing the mean and variance.\n",
+ "\n",
+ "You will find that\n",
+ "\n",
+ "- The mean is unchanged at $ \\mu $, while \n",
+ "- the variance of the portfolio has fallen to $ \\sigma^2 / n $. \n",
+ "\n",
+ "\n",
+ "Diversification reduces risk, as expected.\n",
+ "\n",
+ "But there is a hidden assumption here: the variance of returns is finite.\n",
+ "\n",
+ "If the distribution is heavy-tailed and the variance is infinite, then this\n",
+ "logic is incorrect.\n",
+ "\n",
+ "For example, we saw above that if every $ X_i $ is Cauchy, then so is $ Y_n $.\n",
+ "\n",
+ "This means that diversification doesn’t help at all!"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2403e603",
+ "metadata": {},
+ "source": [
+ "### Fiscal policy\n",
+ "\n",
+ "The heaviness of the tail in the wealth distribution matters for taxation and redistribution policies.\n",
+ "\n",
+ "The same is true for the income distribution.\n",
+ "\n",
+ "For example, the heaviness of the tail of the income distribution helps\n",
+ "determine [how much revenue a given tax policy will raise](https://intro.quantecon.org/mle.html).\n",
+ "\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "147eda94",
+ "metadata": {},
+ "source": [
+ "## Classifying tail properties\n",
+ "\n",
+ "Up until now we have discussed light and heavy tails without any mathematical\n",
+ "definitions.\n",
+ "\n",
+ "Let’s now rectify this.\n",
+ "\n",
+ "We will focus our attention on the right hand tails of\n",
+ "nonnegative random variables and their distributions.\n",
+ "\n",
+ "The definitions for\n",
+ "left hand tails are very similar and we omit them to simplify the exposition.\n",
+ "\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "16fceda4",
+ "metadata": {},
+ "source": [
+ "### Light and heavy tails\n",
+ "\n",
+ "A distribution $ F $ with density $ f $ on $ \\mathbb R_+ $ is called [heavy-tailed](https://en.wikipedia.org/wiki/Heavy-tailed_distribution) if\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\int_0^\\infty \\exp(tx) f(x) dx = \\infty \\; \\text{ for all } t > 0. \\tag{22.4}\n",
+ "$$\n",
+ "\n",
+ "We say that a nonnegative random variable $ X $ is **heavy-tailed** if its density is heavy-tailed.\n",
+ "\n",
+ "This is equivalent to stating that its **moment generating function** $ m(t) :=\n",
+ "\\mathbb E \\exp(t X) $ is infinite for all $ t > 0 $.\n",
+ "\n",
+ "For example, the [log-normal\n",
+ "distribution](https://en.wikipedia.org/wiki/Log-normal_distribution) is\n",
+ "heavy-tailed because its moment generating function is infinite everywhere on\n",
+ "$ (0, \\infty) $.\n",
+ "\n",
+ "The Pareto distribution is also heavy-tailed.\n",
+ "\n",
+ "Less formally, a heavy-tailed distribution is one that is not exponentially bounded (i.e. the tails are heavier than the exponential distribution).\n",
+ "\n",
+ "A distribution $ F $ on $ \\mathbb R_+ $ is called **light-tailed** if it is not heavy-tailed.\n",
+ "\n",
+ "A nonnegative random variable $ X $ is **light-tailed** if its distribution $ F $ is light-tailed.\n",
+ "\n",
+ "For example, every random variable with bounded support is light-tailed. (Why?)\n",
+ "\n",
+ "As another example, if $ X $ has the [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution), with cdf $ F(x) = 1 - \\exp(-\\lambda x) $ for some $ \\lambda > 0 $, then its moment generating function is\n",
+ "\n",
+ "$$\n",
+ "m(t) = \\frac{\\lambda}{\\lambda - t} \\quad \\text{when } t < \\lambda\n",
+ "$$\n",
+ "\n",
+ "In particular, $ m(t) $ is finite whenever $ t < \\lambda $, so $ X $ is light-tailed.\n",
+ "\n",
+ "One can show that if $ X $ is light-tailed, then all of its\n",
+ "[moments](https://en.wikipedia.org/wiki/Moment_%28mathematics%29) are finite.\n",
+ "\n",
+ "Conversely, if some moment is infinite, then $ X $ is heavy-tailed.\n",
+ "\n",
+ "The latter condition is not necessary, however.\n",
+ "\n",
+ "For example, the lognormal distribution is heavy-tailed but every moment is finite."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "038ebb80",
+ "metadata": {},
+ "source": [
+ "## Further reading\n",
+ "\n",
+ "For more on heavy tails in the wealth distribution, see e.g., [[Vilfredo, 1896](https://intro.quantecon.org/zreferences.html#id90)] and [[Benhabib and Bisin, 2018](https://intro.quantecon.org/zreferences.html#id89)].\n",
+ "\n",
+ "For more on heavy tails in the firm size distribution, see e.g., [[Axtell, 2001](https://intro.quantecon.org/zreferences.html#id88)], [[Gabaix, 2016](https://intro.quantecon.org/zreferences.html#id87)].\n",
+ "\n",
+ "For more on heavy tails in the city size distribution, see e.g., [[Rozenfeld *et al.*, 2011](https://intro.quantecon.org/zreferences.html#id84)], [[Gabaix, 2016](https://intro.quantecon.org/zreferences.html#id87)].\n",
+ "\n",
+ "There are other important implications of heavy tails, aside from those\n",
+ "discussed above.\n",
+ "\n",
+ "For example, heavy tails in income and wealth affect productivity growth, business cycles, and political economy.\n",
+ "\n",
+ "For further reading, see, for example, [[Acemoglu and Robinson, 2002](https://intro.quantecon.org/zreferences.html#id83)], [[Glaeser *et al.*, 2003](https://intro.quantecon.org/zreferences.html#id82)], [[Bhandari *et al.*, 2018](https://intro.quantecon.org/zreferences.html#id81)] or [[Ahn *et al.*, 2018](https://intro.quantecon.org/zreferences.html#id80)]."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3577bdfc",
+ "metadata": {},
+ "source": [
+ "## Exercises"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e36f7a41",
+ "metadata": {},
+ "source": [
+ "## Exercise 22.2\n",
+ "\n",
+ "Prove: If $ X $ has a Pareto tail with tail index $ \\alpha $, then\n",
+ "$ \\mathbb E[X^r] = \\infty $ for all $ r \\geq \\alpha $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "81c30d58",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 22.2](https://intro.quantecon.org/#ht_ex2)\n",
+ "\n",
+ "Let $ X $ have a Pareto tail with tail index $ \\alpha $ and let $ F $ be its cdf.\n",
+ "\n",
+ "Fix $ r \\geq \\alpha $.\n",
+ "\n",
+ "In view of [(22.2)](#equation-plrt), we can take positive constants $ b $ and $ \\bar x $ such that\n",
+ "\n",
+ "$$\n",
+ "\\mathbb P\\{X > x\\} \\geq b x^{- \\alpha} \\text{ whenever } x \\geq \\bar x\n",
+ "$$\n",
+ "\n",
+ "But then\n",
+ "\n",
+ "$$\n",
+ "\\mathbb E X^r = r \\int_0^\\infty x^{r-1} \\mathbb P\\{ X > x \\} dx\n",
+ "\\geq\n",
+ "r \\int_0^{\\bar x} x^{r-1} \\mathbb P\\{ X > x \\} dx\n",
+ "+ r \\int_{\\bar x}^\\infty x^{r-1} b x^{-\\alpha} dx.\n",
+ "$$\n",
+ "\n",
+ "We know that $ \\int_{\\bar x}^\\infty x^{r-\\alpha-1} dx = \\infty $ whenever $ r - \\alpha - 1 \\geq -1 $.\n",
+ "\n",
+ "Since $ r \\geq \\alpha $, we have $ \\mathbb E X^r = \\infty $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1f342fd1",
+ "metadata": {},
+ "source": [
+ "## Exercise 22.3\n",
+ "\n",
+ "Repeat exercise 1, but replace the three distributions (two normal, one\n",
+ "Cauchy) with three Pareto distributions using different choices of\n",
+ "$ \\alpha $.\n",
+ "\n",
+ "For $ \\alpha $, try 1.15, 1.5 and 1.75.\n",
+ "\n",
+ "Use `np.random.seed(11)` to set the seed."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1dfda2b8",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 22.3](https://intro.quantecon.org/#ht_ex3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1c746591",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "from scipy.stats import pareto\n",
+ "\n",
+ "np.random.seed(11)\n",
+ "\n",
+ "n = 120\n",
+ "alphas = [1.15, 1.50, 1.75]\n",
+ "\n",
+ "fig, axes = plt.subplots(3, 1, figsize=(6, 8))\n",
+ "\n",
+ "for (a, ax) in zip(alphas, axes):\n",
+ " ax.set_ylim((-5, 50))\n",
+ " data = pareto.rvs(size=n, scale=1, b=a)\n",
+ " ax.plot(list(range(n)), data, linestyle='', marker='o', alpha=0.5, ms=4)\n",
+ " ax.vlines(list(range(n)), 0, data, lw=0.2)\n",
+ " ax.set_title(f\"Pareto draws with $\\\\alpha = {a}$\", fontsize=11)\n",
+ "\n",
+ "plt.subplots_adjust(hspace=0.4)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6553f697",
+ "metadata": {},
+ "source": [
+ "## Exercise 22.4\n",
+ "\n",
+ "There is an ongoing argument about whether the firm size distribution should\n",
+ "be modeled as a Pareto distribution or a lognormal distribution (see, e.g.,\n",
+ "[[Fujiwara *et al.*, 2004](https://intro.quantecon.org/zreferences.html#id73)], [[Kondo *et al.*, 2018](https://intro.quantecon.org/zreferences.html#id71)] or [[Schluter and Trede, 2019](https://intro.quantecon.org/zreferences.html#id72)]).\n",
+ "\n",
+ "This sounds esoteric but has real implications for a variety of economic\n",
+ "phenomena.\n",
+ "\n",
+ "To illustrate this fact in a simple way, let us consider an economy with\n",
+ "100,000 firms, an interest rate of `r = 0.05` and a corporate tax rate of\n",
+ "15%.\n",
+ "\n",
+ "Your task is to estimate the present discounted value of projected corporate\n",
+ "tax revenue over the next 10 years.\n",
+ "\n",
+ "Because we are forecasting, we need a model.\n",
+ "\n",
+ "We will suppose that\n",
+ "\n",
+ "1. the number of firms and the firm size distribution (measured in profits) remain fixed and \n",
+ "1. the firm size distribution is either lognormal or Pareto. \n",
+ "\n",
+ "\n",
+ "Present discounted value of tax revenue will be estimated by\n",
+ "\n",
+ "1. generating 100,000 draws of firm profit from the firm size distribution, \n",
+ "1. multiplying by the tax rate, and \n",
+ "1. summing the results with discounting to obtain present value. \n",
+ "\n",
+ "\n",
+ "The Pareto distribution is assumed to take the form [(22.1)](#equation-pareto) with $ \\bar x = 1 $ and $ \\alpha = 1.05 $.\n",
+ "\n",
+ "(The value of the tail index $ \\alpha $ is plausible given the data [[Gabaix, 2016](https://intro.quantecon.org/zreferences.html#id87)].)\n",
+ "\n",
+ "To make the lognormal option as similar as possible to the Pareto option, choose\n",
+ "its parameters such that the mean and median of both distributions are the same.\n",
+ "\n",
+ "Note that, for each distribution, your estimate of tax revenue will be random\n",
+ "because it is based on a finite number of draws.\n",
+ "\n",
+ "To take this into account, generate 100 replications (evaluations of tax revenue)\n",
+ "for each of the two distributions and compare the two samples by\n",
+ "\n",
+ "- producing a [violin plot](https://en.wikipedia.org/wiki/Violin_plot) visualizing the two samples side-by-side and \n",
+ "- printing the mean and standard deviation of both samples. \n",
+ "\n",
+ "\n",
+ "For the seed use `np.random.seed(1234)`.\n",
+ "\n",
+ "What differences do you observe?\n",
+ "\n",
+ "(Note: a better approach to this problem would be to model firm dynamics and\n",
+ "try to track individual firms given the current distribution. We will discuss\n",
+ "firm dynamics in later lectures.)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4db8edb0",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 22.4](https://intro.quantecon.org/#ht_ex5)\n",
+ "\n",
+ "To do the exercise, we need to choose the parameters $ \\mu $\n",
+ "and $ \\sigma $ of the lognormal distribution to match the mean and median\n",
+ "of the Pareto distribution.\n",
+ "\n",
+ "Here we understand the lognormal distribution as that of the random variable\n",
+ "$ \\exp(\\mu + \\sigma Z) $ when $ Z $ is standard normal.\n",
+ "\n",
+ "The mean and median of the Pareto distribution [(22.1)](#equation-pareto) with\n",
+ "$ \\bar x = 1 $ are\n",
+ "\n",
+ "$$\n",
+ "\\text{mean } = \\frac{\\alpha}{\\alpha - 1}\n",
+ "\\quad \\text{and} \\quad\n",
+ "\\text{median } = 2^{1/\\alpha}\n",
+ "$$\n",
+ "\n",
+ "Using the corresponding expressions for the lognormal distribution leads us to\n",
+ "the equations\n",
+ "\n",
+ "$$\n",
+ "\\frac{\\alpha}{\\alpha - 1} = \\exp(\\mu + \\sigma^2/2)\n",
+ "\\quad \\text{and} \\quad\n",
+ "2^{1/\\alpha} = \\exp(\\mu)\n",
+ "$$\n",
+ "\n",
+ "which we solve for $ \\mu $ and $ \\sigma $ given $ \\alpha = 1.05 $.\n",
+ "\n",
+ "Here is the code that generates the two samples, produces the violin plot and\n",
+ "prints the mean and standard deviation of the two samples."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4c5547a8",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "num_firms = 100_000\n",
+ "num_years = 10\n",
+ "tax_rate = 0.15\n",
+ "r = 0.05\n",
+ "\n",
+ "β = 1 / (1 + r) # discount factor\n",
+ "\n",
+ "x_bar = 1.0\n",
+ "α = 1.05\n",
+ "\n",
+ "def pareto_rvs(n):\n",
+ " \"Uses a standard method to generate Pareto draws.\"\n",
+ " u = np.random.uniform(size=n)\n",
+ " y = x_bar / (u**(1/α))\n",
+ " return y"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b2e2f656",
+ "metadata": {},
+ "source": [
+ "Let’s compute the lognormal parameters:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6eb04911",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "μ = np.log(2) / α\n",
+ "σ_sq = 2 * (np.log(α/(α - 1)) - np.log(2)/α)\n",
+ "σ = np.sqrt(σ_sq)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "473fecf2",
+ "metadata": {},
+ "source": [
+ "Here’s a function to compute a single estimate of tax revenue for a particular\n",
+ "choice of distribution `dist`."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b9298ebb",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def tax_rev(dist):\n",
+ " tax_raised = 0\n",
+ " for t in range(num_years):\n",
+ " if dist == 'pareto':\n",
+ " π = pareto_rvs(num_firms)\n",
+ " else:\n",
+ " π = np.exp(μ + σ * np.random.randn(num_firms))\n",
+ " tax_raised += β**t * np.sum(π * tax_rate)\n",
+ " return tax_raised"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f8d4463f",
+ "metadata": {},
+ "source": [
+ "Now let’s generate the violin plot."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "90c0249a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "num_reps = 100\n",
+ "np.random.seed(1234)\n",
+ "\n",
+ "tax_rev_lognorm = np.empty(num_reps)\n",
+ "tax_rev_pareto = np.empty(num_reps)\n",
+ "\n",
+ "for i in range(num_reps):\n",
+ " tax_rev_pareto[i] = tax_rev('pareto')\n",
+ " tax_rev_lognorm[i] = tax_rev('lognorm')\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "\n",
+ "data = tax_rev_pareto, tax_rev_lognorm\n",
+ "\n",
+ "ax.violinplot(data)\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "32abad1b",
+ "metadata": {},
+ "source": [
+ "Finally, let’s print the means and standard deviations."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "cd295b96",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "tax_rev_pareto.mean(), tax_rev_pareto.std()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "078a1e1c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "tax_rev_lognorm.mean(), tax_rev_lognorm.std()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e0c2a929",
+ "metadata": {},
+ "source": [
+ "Looking at the output of the code, our main conclusion is that the Pareto\n",
+ "assumption leads to a lower mean and greater dispersion."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5d09787c",
+ "metadata": {},
+ "source": [
+ "## Exercise 22.5\n",
+ "\n",
+ "The [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29) of the Cauchy distribution is\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\phi(t) = \\mathbb E e^{itX} = \\int e^{i t x} f(x) dx = e^{-|t|} \\tag{22.5}\n",
+ "$$\n",
+ "\n",
+ "Prove that the sample mean $ \\bar X_n $ of $ n $ independent draws $ X_1, \\ldots,\n",
+ "X_n $ from the Cauchy distribution has the same characteristic function as\n",
+ "$ X_1 $.\n",
+ "\n",
+ "(This means that the sample mean never converges.)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "431bc4e4",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 22.5](https://intro.quantecon.org/#ht_ex_cauchy)\n",
+ "\n",
+ "By independence, the characteristic function of the sample mean becomes\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ " \\mathbb E e^{i t \\bar X_n }\n",
+ " & = \\mathbb E \\exp \\left\\{ i \\frac{t}{n} \\sum_{j=1}^n X_j \\right\\}\n",
+ " \\\\\n",
+ " & = \\mathbb E \\prod_{j=1}^n \\exp \\left\\{ i \\frac{t}{n} X_j \\right\\}\n",
+ " \\\\\n",
+ " & = \\prod_{j=1}^n \\mathbb E \\exp \\left\\{ i \\frac{t}{n} X_j \\right\\}\n",
+ " = [\\phi(t/n)]^n\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "In view of [(22.5)](#equation-lln-cch), this is just $ e^{-|t|} $.\n",
+ "\n",
+ "Thus, in the case of the Cauchy distribution, the sample mean itself has the very same Cauchy distribution, regardless of $ n $!"
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476280.9830956,
+ "filename": "heavy_tails.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Heavy-Tailed Distributions"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/inequality.ipynb b/_notebooks/inequality.ipynb
new file mode 100644
index 000000000..8ef5fdd38
--- /dev/null
+++ b/_notebooks/inequality.ipynb
@@ -0,0 +1,1786 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "b754a180",
+ "metadata": {},
+ "source": [
+ "# Income and Wealth Inequality"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2318b5ef",
+ "metadata": {},
+ "source": [
+ "## Overview\n",
+ "\n",
+ "In the lecture [Long-Run Growth](https://intro.quantecon.org/long_run_growth.html) we studied how GDP per capita has changed\n",
+ "for certain countries and regions.\n",
+ "\n",
+ "Per capita GDP is important because it gives us an idea of average income for\n",
+ "households in a given country.\n",
+ "\n",
+ "However, when we study income and wealth, averages are only part of the story."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "add485b8",
+ "metadata": {},
+ "source": [
+ "## \n",
+ "\n",
+ "For example, imagine two societies, each with one million people, where\n",
+ "\n",
+ "- in the first society, the yearly income of one man is \\$100,000,000 and the income of the\n",
+ " others are zero \n",
+ "- in the second society, the yearly income of everyone is \\$100 \n",
+ "\n",
+ "\n",
+ "These countries have the same income per capita (average income is \\$100) but the lives of the people will be very different (e.g., almost everyone in the first society is\n",
+ "starving, even though one person is fabulously rich).\n",
+ "\n",
+ "The example above suggests that we should go beyond simple averages when we study income and wealth.\n",
+ "\n",
+ "This leads us to the topic of economic inequality, which examines how income and wealth (and other quantities) are distributed across a population.\n",
+ "\n",
+ "In this lecture we study inequality, beginning with measures of inequality and\n",
+ "then applying them to wealth and income data from the US and other countries."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4c121ffe",
+ "metadata": {},
+ "source": [
+ "### Some history\n",
+ "\n",
+ "Many historians argue that inequality played a role in the fall of the Roman Republic (see, e.g., [[Levitt, 2019](https://intro.quantecon.org/zreferences.html#id7)]).\n",
+ "\n",
+ "Following the defeat of Carthage and the invasion of Spain, money flowed into\n",
+ "Rome from across the empire, greatly enriched those in power.\n",
+ "\n",
+ "Meanwhile, ordinary citizens were taken from their farms to fight for long\n",
+ "periods, diminishing their wealth.\n",
+ "\n",
+ "The resulting growth in inequality was a driving factor behind political turmoil that shook the foundations of the republic.\n",
+ "\n",
+ "Eventually, the Roman Republic gave way to a series of dictatorships, starting with [Octavian](https://en.wikipedia.org/wiki/Augustus) (Augustus) in 27 BCE.\n",
+ "\n",
+ "This history tells us that inequality matters, in the sense that it can drive major world events.\n",
+ "\n",
+ "There are other reasons that inequality might matter, such as how it affects\n",
+ "human welfare.\n",
+ "\n",
+ "With this motivation, let us start to think about what inequality is and how we\n",
+ "can quantify and analyze it."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "64917730",
+ "metadata": {},
+ "source": [
+ "### Measurement\n",
+ "\n",
+ "In politics and popular media, the word “inequality” is often used quite loosely, without any firm definition.\n",
+ "\n",
+ "To bring a scientific perspective to the topic of inequality we must start with careful definitions.\n",
+ "\n",
+ "Hence we begin by discussing ways that inequality can be measured in economic research.\n",
+ "\n",
+ "We will need to install the following packages"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "04d51ea7",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "!pip install wbgapi plotly"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e17c3a12",
+ "metadata": {},
+ "source": [
+ "We will also use the following imports."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7120d35a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import pandas as pd\n",
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "import random as rd\n",
+ "import wbgapi as wb\n",
+ "import plotly.express as px"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fa78749c",
+ "metadata": {},
+ "source": [
+ "## The Lorenz curve\n",
+ "\n",
+ "One popular measure of inequality is the Lorenz curve.\n",
+ "\n",
+ "In this section we define the Lorenz curve and examine its properties."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4e6be000",
+ "metadata": {},
+ "source": [
+ "### Definition\n",
+ "\n",
+ "The Lorenz curve takes a sample $ w_1, \\ldots, w_n $ and produces a curve $ L $.\n",
+ "\n",
+ "We suppose that the sample has been sorted from smallest to largest.\n",
+ "\n",
+ "To aid our interpretation, suppose that we are measuring wealth\n",
+ "\n",
+ "- $ w_1 $ is the wealth of the poorest member of the population, and \n",
+ "- $ w_n $ is the wealth of the richest member of the population. \n",
+ "\n",
+ "\n",
+ "The curve $ L $ is just a function $ y = L(x) $ that we can plot and interpret.\n",
+ "\n",
+ "To create it we first generate data points $ (x_i, y_i) $ according to"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dcd79ac1",
+ "metadata": {},
+ "source": [
+ "### \n",
+ "\n",
+ "$$\n",
+ "x_i = \\frac{i}{n},\n",
+ "\\qquad\n",
+ "y_i = \\frac{\\sum_{j \\leq i} w_j}{\\sum_{j \\leq n} w_j},\n",
+ "\\qquad i = 1, \\ldots, n\n",
+ "$$\n",
+ "\n",
+ "Now the Lorenz curve $ L $ is formed from these data points using interpolation.\n",
+ "\n",
+ "If we use a line plot in `matplotlib`, the interpolation will be done for us.\n",
+ "\n",
+ "The meaning of the statement $ y = L(x) $ is that the lowest $ (100\n",
+ "\\times x) $% of people have $ (100 \\times y) $% of all wealth.\n",
+ "\n",
+ "- if $ x=0.5 $ and $ y=0.1 $, then the bottom 50% of the population\n",
+ " owns 10% of the wealth. \n",
+ "\n",
+ "\n",
+ "In the discussion above we focused on wealth but the same ideas apply to\n",
+ "income, consumption, etc."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "56d459b6",
+ "metadata": {},
+ "source": [
+ "### Lorenz curves of simulated data\n",
+ "\n",
+ "Let’s look at some examples and try to build understanding.\n",
+ "\n",
+ "First let us construct a `lorenz_curve` function that we can\n",
+ "use in our simulations below.\n",
+ "\n",
+ "It is useful to construct a function that translates an array of\n",
+ "income or wealth data into the cumulative share\n",
+ "of individuals (or households) and the cumulative share of income (or wealth)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9cc5dc71",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def lorenz_curve(y):\n",
+ " \"\"\"\n",
+ " Calculates the Lorenz Curve, a graphical representation of\n",
+ " the distribution of income or wealth.\n",
+ "\n",
+ " It returns the cumulative share of people (x-axis) and\n",
+ " the cumulative share of income earned.\n",
+ "\n",
+ " Parameters\n",
+ " ----------\n",
+ " y : array_like(float or int, ndim=1)\n",
+ " Array of income/wealth for each individual.\n",
+ " Unordered or ordered is fine.\n",
+ "\n",
+ " Returns\n",
+ " -------\n",
+ " cum_people : array_like(float, ndim=1)\n",
+ " Cumulative share of people for each person index (i/n)\n",
+ " cum_income : array_like(float, ndim=1)\n",
+ " Cumulative share of income for each person index\n",
+ "\n",
+ "\n",
+ " References\n",
+ " ----------\n",
+ " .. [1] https://en.wikipedia.org/wiki/Lorenz_curve\n",
+ "\n",
+ " Examples\n",
+ " --------\n",
+ " >>> a_val, n = 3, 10_000\n",
+ " >>> y = np.random.pareto(a_val, size=n)\n",
+ " >>> f_vals, l_vals = lorenz(y)\n",
+ "\n",
+ " \"\"\"\n",
+ "\n",
+ " n = len(y)\n",
+ " y = np.sort(y)\n",
+ " s = np.zeros(n + 1)\n",
+ " s[1:] = np.cumsum(y)\n",
+ " cum_people = np.zeros(n + 1)\n",
+ " cum_income = np.zeros(n + 1)\n",
+ " for i in range(1, n + 1):\n",
+ " cum_people[i] = i / n\n",
+ " cum_income[i] = s[i] / s[n]\n",
+ " return cum_people, cum_income"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2237e8fa",
+ "metadata": {},
+ "source": [
+ "In the next figure, we generate $ n=2000 $ draws from a lognormal\n",
+ "distribution and treat these draws as our population.\n",
+ "\n",
+ "The straight 45-degree line ($ x=L(x) $ for all $ x $) corresponds to perfect equality.\n",
+ "\n",
+ "The log-normal draws produce a less equal distribution.\n",
+ "\n",
+ "For example, if we imagine these draws as being observations of wealth across\n",
+ "a sample of households, then the dashed lines show that the bottom 80% of\n",
+ "households own just over 40% of total wealth."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8e66f1e4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "n = 2000\n",
+ "sample = np.exp(np.random.randn(n))\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "\n",
+ "f_vals, l_vals = lorenz_curve(sample)\n",
+ "ax.plot(f_vals, l_vals, label=f'lognormal sample', lw=2)\n",
+ "ax.plot(f_vals, f_vals, label='equality', lw=2)\n",
+ "\n",
+ "ax.vlines([0.8], [0.0], [0.43], alpha=0.5, colors='k', ls='--')\n",
+ "ax.hlines([0.43], [0], [0.8], alpha=0.5, colors='k', ls='--')\n",
+ "ax.set_xlim((0, 1))\n",
+ "ax.set_xlabel(\"share of households\")\n",
+ "ax.set_ylim((0, 1))\n",
+ "ax.set_ylabel(\"share of wealth\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3df9dc0c",
+ "metadata": {},
+ "source": [
+ "### Lorenz curves for US data\n",
+ "\n",
+ "Next let’s look at US data for both income and wealth.\n",
+ "\n",
+ "\n",
+ "\n",
+ "The following code block imports a subset of the dataset `SCF_plus` for 2016,\n",
+ "which is derived from the [Survey of Consumer Finances](https://en.wikipedia.org/wiki/Survey_of_Consumer_Finances) (SCF)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "02d82bf2",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "url = 'https://github.com/QuantEcon/high_dim_data/raw/main/SCF_plus/SCF_plus_mini.csv'\n",
+ "df = pd.read_csv(url)\n",
+ "df_income_wealth = df.dropna()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "20e5a2d0",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_income_wealth.head(n=5)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "02018d3d",
+ "metadata": {},
+ "source": [
+ "The next code block uses data stored in dataframe `df_income_wealth` to generate the Lorenz curves.\n",
+ "\n",
+ "(The code is somewhat complex because we need to adjust the data according to\n",
+ "population weights supplied by the SCF.)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ae93ec3e",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df = df_income_wealth \n",
+ "\n",
+ "varlist = ['n_wealth', # net wealth \n",
+ " 't_income', # total income\n",
+ " 'l_income'] # labor income\n",
+ "\n",
+ "years = df.year.unique()\n",
+ "\n",
+ "# Create lists to store Lorenz data\n",
+ "\n",
+ "F_vals, L_vals = [], []\n",
+ "\n",
+ "for var in varlist:\n",
+ " # create lists to store Lorenz curve data\n",
+ " f_vals = []\n",
+ " l_vals = []\n",
+ " for year in years:\n",
+ "\n",
+ " # Repeat the observations according to their weights\n",
+ " counts = list(round(df[df['year'] == year]['weights'] )) \n",
+ " y = df[df['year'] == year][var].repeat(counts)\n",
+ " y = np.asarray(y)\n",
+ " \n",
+ " # Shuffle the sequence to improve the plot\n",
+ " rd.shuffle(y) \n",
+ " \n",
+ " # calculate and store Lorenz curve data\n",
+ " f_val, l_val = lorenz_curve(y)\n",
+ " f_vals.append(f_val)\n",
+ " l_vals.append(l_val)\n",
+ " \n",
+ " F_vals.append(f_vals)\n",
+ " L_vals.append(l_vals)\n",
+ "\n",
+ "f_vals_nw, f_vals_ti, f_vals_li = F_vals\n",
+ "l_vals_nw, l_vals_ti, l_vals_li = L_vals"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ef9f5d24",
+ "metadata": {},
+ "source": [
+ "Now we plot Lorenz curves for net wealth, total income and labor income in the\n",
+ "US in 2016.\n",
+ "\n",
+ "Total income is the sum of households’ all income sources, including labor income but excluding capital gains.\n",
+ "\n",
+ "(All income measures are pre-tax.)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e69412c1",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax.plot(f_vals_nw[-1], l_vals_nw[-1], label=f'net wealth')\n",
+ "ax.plot(f_vals_ti[-1], l_vals_ti[-1], label=f'total income')\n",
+ "ax.plot(f_vals_li[-1], l_vals_li[-1], label=f'labor income')\n",
+ "ax.plot(f_vals_nw[-1], f_vals_nw[-1], label=f'equality')\n",
+ "ax.set_xlabel(\"share of households\")\n",
+ "ax.set_ylabel(\"share of income/wealth\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8ce8f8d3",
+ "metadata": {},
+ "source": [
+ "One key finding from this figure is that wealth inequality is more extreme than income inequality."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a15f459d",
+ "metadata": {},
+ "source": [
+ "## The Gini coefficient\n",
+ "\n",
+ "The Lorenz curve provides a visual representation of inequality in a distribution.\n",
+ "\n",
+ "Another way to study income and wealth inequality is via the Gini coefficient.\n",
+ "\n",
+ "In this section we discuss the Gini coefficient and its relationship to the Lorenz curve."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "974c1924",
+ "metadata": {},
+ "source": [
+ "### Definition\n",
+ "\n",
+ "As before, suppose that the sample $ w_1, \\ldots, w_n $ has been sorted from smallest to largest.\n",
+ "\n",
+ "The Gini coefficient is defined for the sample above as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "43937328",
+ "metadata": {},
+ "source": [
+ "### \n",
+ "\n",
+ "$$\n",
+ "G :=\n",
+ "\\frac{\\sum_{i=1}^n \\sum_{j = 1}^n |w_j - w_i|}\n",
+ " {2n\\sum_{i=1}^n w_i}.\n",
+ "$$\n",
+ "\n",
+ "The Gini coefficient is closely related to the Lorenz curve.\n",
+ "\n",
+ "In fact, it can be shown that its value is twice the area between the line of\n",
+ "equality and the Lorenz curve (e.g., the shaded area in Fig. 6.3).\n",
+ "\n",
+ "The idea is that $ G=0 $ indicates complete equality, while $ G=1 $ indicates complete inequality."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "aa44aac7",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "f_vals, l_vals = lorenz_curve(sample)\n",
+ "ax.plot(f_vals, l_vals, label=f'lognormal sample', lw=2)\n",
+ "ax.plot(f_vals, f_vals, label='equality', lw=2)\n",
+ "ax.fill_between(f_vals, l_vals, f_vals, alpha=0.06)\n",
+ "ax.set_ylim((0, 1))\n",
+ "ax.set_xlim((0, 1))\n",
+ "ax.text(0.04, 0.5, r'$G = 2 \\times$ shaded area')\n",
+ "ax.set_xlabel(\"share of households (%)\")\n",
+ "ax.set_ylabel(\"share of wealth (%)\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "152c3b43",
+ "metadata": {},
+ "source": [
+ "In fact the Gini coefficient can also be expressed as\n",
+ "\n",
+ "$$\n",
+ "G = \\frac{A}{A+B}\n",
+ "$$\n",
+ "\n",
+ "where $ A $ is the area between the 45-degree line of\n",
+ "perfect equality and the Lorenz curve, while $ B $ is the area below the Lorenze curve – see Fig. 6.4."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9e819095",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "f_vals, l_vals = lorenz_curve(sample)\n",
+ "ax.plot(f_vals, l_vals, label='lognormal sample', lw=2)\n",
+ "ax.plot(f_vals, f_vals, label='equality', lw=2)\n",
+ "ax.fill_between(f_vals, l_vals, f_vals, alpha=0.06)\n",
+ "ax.fill_between(f_vals, l_vals, np.zeros_like(f_vals), alpha=0.06)\n",
+ "ax.set_ylim((0, 1))\n",
+ "ax.set_xlim((0, 1))\n",
+ "ax.text(0.55, 0.4, 'A')\n",
+ "ax.text(0.75, 0.15, 'B')\n",
+ "ax.set_xlabel(\"share of households\")\n",
+ "ax.set_ylabel(\"share of wealth\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "27176342",
+ "metadata": {},
+ "source": [
+ "The World in Data project has a [graphical exploration of the Lorenz curve and the Gini coefficient](https://ourworldindata.org/what-is-the-gini-coefficient)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7a4d6214",
+ "metadata": {},
+ "source": [
+ "### Gini coefficient of simulated data\n",
+ "\n",
+ "Let’s examine the Gini coefficient in some simulations.\n",
+ "\n",
+ "The code below computes the Gini coefficient from a sample.\n",
+ "\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1b52bda3",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def gini_coefficient(y):\n",
+ " r\"\"\"\n",
+ " Implements the Gini inequality index\n",
+ "\n",
+ " Parameters\n",
+ " ----------\n",
+ " y : array_like(float)\n",
+ " Array of income/wealth for each individual.\n",
+ " Ordered or unordered is fine\n",
+ "\n",
+ " Returns\n",
+ " -------\n",
+ " Gini index: float\n",
+ " The gini index describing the inequality of the array of income/wealth\n",
+ "\n",
+ " References\n",
+ " ----------\n",
+ "\n",
+ " https://en.wikipedia.org/wiki/Gini_coefficient\n",
+ " \"\"\"\n",
+ " n = len(y)\n",
+ " i_sum = np.zeros(n)\n",
+ " for i in range(n):\n",
+ " for j in range(n):\n",
+ " i_sum[i] += abs(y[i] - y[j])\n",
+ " return np.sum(i_sum) / (2 * n * np.sum(y))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "68ce1065",
+ "metadata": {},
+ "source": [
+ "Now we can compute the Gini coefficients for five different populations.\n",
+ "\n",
+ "Each of these populations is generated by drawing from a\n",
+ "lognormal distribution with parameters $ \\mu $ (mean) and $ \\sigma $ (standard deviation).\n",
+ "\n",
+ "To create the five populations, we vary $ \\sigma $ over a grid of length $ 5 $\n",
+ "between $ 0.2 $ and $ 4 $.\n",
+ "\n",
+ "In each case we set $ \\mu = - \\sigma^2 / 2 $.\n",
+ "\n",
+ "This implies that the mean of the distribution does not change with $ \\sigma $.\n",
+ "\n",
+ "You can check this by looking up the expression for the mean of a lognormal\n",
+ "distribution."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "445318cd",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "%%time\n",
+ "k = 5\n",
+ "σ_vals = np.linspace(0.2, 4, k)\n",
+ "n = 2_000\n",
+ "\n",
+ "ginis = []\n",
+ "\n",
+ "for σ in σ_vals:\n",
+ " μ = -σ**2 / 2\n",
+ " y = np.exp(μ + σ * np.random.randn(n))\n",
+ " ginis.append(gini_coefficient(y))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b9e6be03",
+ "metadata": {},
+ "source": [
+ "Let’s build a function that returns a figure (so that we can use it later in the lecture)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "15628c65",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def plot_inequality_measures(x, y, legend, xlabel, ylabel):\n",
+ " fig, ax = plt.subplots()\n",
+ " ax.plot(x, y, marker='o', label=legend)\n",
+ " ax.set_xlabel(xlabel)\n",
+ " ax.set_ylabel(ylabel)\n",
+ " ax.legend()\n",
+ " return fig, ax"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4e1531f5",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fix, ax = plot_inequality_measures(σ_vals, \n",
+ " ginis, \n",
+ " 'simulated', \n",
+ " r'$\\sigma$', \n",
+ " 'Gini coefficients')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "959e8382",
+ "metadata": {},
+ "source": [
+ "The plots show that inequality rises with $ \\sigma $, according to the Gini\n",
+ "coefficient."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f36d27bc",
+ "metadata": {},
+ "source": [
+ "### Gini coefficient for income (US data)\n",
+ "\n",
+ "Let’s look at the Gini coefficient for the distribution of income in the US.\n",
+ "\n",
+ "We will get pre-computed Gini coefficients (based on income) from the World Bank using the [wbgapi](https://blogs.worldbank.org/opendata/introducing-wbgapi-new-python-package-accessing-world-bank-data).\n",
+ "\n",
+ "Let’s use the `wbgapi` package we imported earlier to search the World Bank data for Gini to find the Series ID."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6d4bd54a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "wb.search(\"gini\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "96c50ac3",
+ "metadata": {},
+ "source": [
+ "We now know the series ID is `SI.POV.GINI`.\n",
+ "\n",
+ "(Another way to find the series ID is to use the [World Bank data portal](https://data.worldbank.org) and then use `wbgapi` to fetch the data.)\n",
+ "\n",
+ "To get a quick overview, let’s histogram Gini coefficients across all countries and all years in the World Bank dataset."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "eff49977",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Fetch gini data for all countries\n",
+ "gini_all = wb.data.DataFrame(\"SI.POV.GINI\")\n",
+ "# remove 'YR' in index and convert to integer\n",
+ "gini_all.columns = gini_all.columns.map(lambda x: int(x.replace('YR',''))) \n",
+ "\n",
+ "# Create a long series with a multi-index of the data to get global min and max values\n",
+ "gini_all = gini_all.unstack(level='economy').dropna()\n",
+ "\n",
+ "# Build a histogram\n",
+ "ax = gini_all.plot(kind=\"hist\", bins=20)\n",
+ "ax.set_xlabel(\"Gini coefficient\")\n",
+ "ax.set_ylabel(\"frequency\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9a862f5b",
+ "metadata": {},
+ "source": [
+ "We can see in Fig. 6.6 that across 50 years of data and all countries the measure varies between 20 and 65.\n",
+ "\n",
+ "Let us fetch the data `DataFrame` for the USA."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "bd839c86",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data = wb.data.DataFrame(\"SI.POV.GINI\", \"USA\")\n",
+ "data.head(n=5)\n",
+ "# remove 'YR' in index and convert to integer\n",
+ "data.columns = data.columns.map(lambda x: int(x.replace('YR','')))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ec4eabff",
+ "metadata": {},
+ "source": [
+ "(This package often returns data with year information contained in the columns. This is not always convenient for simple plotting with pandas so it can be useful to transpose the results before plotting.)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ae5926ed",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data = data.T # Obtain years as rows\n",
+ "data_usa = data['USA'] # pd.Series of US data"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6df49946",
+ "metadata": {},
+ "source": [
+ "Let us take a look at the data for the US."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "891f5738",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax = data_usa.plot(ax=ax)\n",
+ "ax.set_ylim(data_usa.min()-1, data_usa.max()+1)\n",
+ "ax.set_ylabel(\"Gini coefficient (income)\")\n",
+ "ax.set_xlabel(\"year\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c5b4d6e8",
+ "metadata": {},
+ "source": [
+ "As can be seen in Fig. 6.7, the income Gini\n",
+ "trended upward from 1980 to 2020 and then dropped following at the start of the COVID pandemic.\n",
+ "\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "54a96a21",
+ "metadata": {},
+ "source": [
+ "### Gini coefficient for wealth\n",
+ "\n",
+ "In the previous section we looked at the Gini coefficient for income, focusing on using US data.\n",
+ "\n",
+ "Now let’s look at the Gini coefficient for the distribution of wealth.\n",
+ "\n",
+ "We will use US data from the [Survey of Consumer Finances](#data-survey-consumer-finance)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3d797eb4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_income_wealth.year.describe()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "29e756ad",
+ "metadata": {},
+ "source": [
+ "[This notebook](https://github.com/QuantEcon/lecture-python-intro/tree/main/lectures/_static/lecture_specific/inequality/data.ipynb) can be used to compute this information over the full dataset."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f0136da6",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data_url = 'https://github.com/QuantEcon/lecture-python-intro/raw/main/lectures/_static/lecture_specific/inequality/usa-gini-nwealth-tincome-lincome.csv'\n",
+ "ginis = pd.read_csv(data_url, index_col='year')\n",
+ "ginis.head(n=5)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0f5967a4",
+ "metadata": {},
+ "source": [
+ "Let’s plot the Gini coefficients for net wealth."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ac0b2f13",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax.plot(years, ginis[\"n_wealth\"], marker='o')\n",
+ "ax.set_xlabel(\"year\")\n",
+ "ax.set_ylabel(\"Gini coefficient\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "03bee88e",
+ "metadata": {},
+ "source": [
+ "The time series for the wealth Gini exhibits a U-shape, falling until the early\n",
+ "1980s and then increasing rapidly.\n",
+ "\n",
+ "One possibility is that this change is mainly driven by technology.\n",
+ "\n",
+ "However, we will see below that not all advanced economies experienced similar growth of inequality."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5b15b627",
+ "metadata": {},
+ "source": [
+ "### Cross-country comparisons of income inequality\n",
+ "\n",
+ "Earlier in this lecture we used `wbgapi` to get Gini data across many countries\n",
+ "and saved it in a variable called `gini_all`\n",
+ "\n",
+ "In this section we will use this data to compare several advanced economies, and\n",
+ "to look at the evolution in their respective income Ginis."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9b9104a4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data = gini_all.unstack()\n",
+ "data.columns"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "91f5688d",
+ "metadata": {},
+ "source": [
+ "There are 167 countries represented in this dataset.\n",
+ "\n",
+ "Let us compare three advanced economies: the US, the UK, and Norway"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "44c9f788",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "ax = data[['USA','GBR', 'NOR']].plot()\n",
+ "ax.set_xlabel('year')\n",
+ "ax.set_ylabel('Gini coefficient')\n",
+ "ax.legend(title=\"\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6ac3cf94",
+ "metadata": {},
+ "source": [
+ "We see that Norway has a shorter time series.\n",
+ "\n",
+ "Let us take a closer look at the underlying data and see if we can rectify this."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "01a22a4b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data[['NOR']].dropna().head(n=5)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d7017c58",
+ "metadata": {},
+ "source": [
+ "The data for Norway in this dataset goes back to 1979 but there are gaps in the time series and matplotlib is not showing those data points.\n",
+ "\n",
+ "We can use the `.ffill()` method to copy and bring forward the last known value in a series to fill in these gaps"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fb87aec7",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data['NOR'] = data['NOR'].ffill()\n",
+ "ax = data[['USA','GBR', 'NOR']].plot()\n",
+ "ax.set_xlabel('year')\n",
+ "ax.set_ylabel('Gini coefficient')\n",
+ "ax.legend(title=\"\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "68bca724",
+ "metadata": {},
+ "source": [
+ "From this plot we can observe that the US has a higher Gini coefficient (i.e.\n",
+ "higher income inequality) when compared to the UK and Norway.\n",
+ "\n",
+ "Norway has the lowest Gini coefficient over the three economies and, moreover,\n",
+ "the Gini coefficient shows no upward trend."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d98c126d",
+ "metadata": {},
+ "source": [
+ "### Gini Coefficient and GDP per capita (over time)\n",
+ "\n",
+ "We can also look at how the Gini coefficient compares with GDP per capita (over time).\n",
+ "\n",
+ "Let’s take another look at the US, Norway, and the UK."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6cdc687c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "countries = ['USA', 'NOR', 'GBR']\n",
+ "gdppc = wb.data.DataFrame(\"NY.GDP.PCAP.KD\", countries)\n",
+ "# remove 'YR' in index and convert to integer\n",
+ "gdppc.columns = gdppc.columns.map(lambda x: int(x.replace('YR',''))) \n",
+ "gdppc = gdppc.T"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9e5f550c",
+ "metadata": {},
+ "source": [
+ "We can rearrange the data so that we can plot GDP per capita and the Gini coefficient across years"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "70ebaf36",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plot_data = pd.DataFrame(data[countries].unstack())\n",
+ "plot_data.index.names = ['country', 'year']\n",
+ "plot_data.columns = ['gini']"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6f3078ec",
+ "metadata": {},
+ "source": [
+ "Now we can get the GDP per capita data into a shape that can be merged with `plot_data`"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3880237c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "pgdppc = pd.DataFrame(gdppc.unstack())\n",
+ "pgdppc.index.names = ['country', 'year']\n",
+ "pgdppc.columns = ['gdppc']\n",
+ "plot_data = plot_data.merge(pgdppc, left_index=True, right_index=True)\n",
+ "plot_data.reset_index(inplace=True)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e41fc8b0",
+ "metadata": {},
+ "source": [
+ "Now we use Plotly to build a plot with GDP per capita on the y-axis and the Gini coefficient on the x-axis."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9cb35f44",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "min_year = plot_data.year.min()\n",
+ "max_year = plot_data.year.max()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fe5cdc4c",
+ "metadata": {},
+ "source": [
+ "The time series for all three countries start and stop in different years.\n",
+ "\n",
+ "We will add a year mask to the data to improve clarity in the chart including the different end years associated with each country’s time series."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "82f63447",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "labels = [1979, 1986, 1991, 1995, 2000, 2020, 2021, 2022] + \\\n",
+ " list(range(min_year,max_year,5))\n",
+ "plot_data.year = plot_data.year.map(lambda x: x if x in labels else None)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2531492e",
+ "metadata": {},
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "03847dfb",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig = px.line(plot_data, \n",
+ " x = \"gini\", \n",
+ " y = \"gdppc\", \n",
+ " color = \"country\", \n",
+ " text = \"year\", \n",
+ " height = 800,\n",
+ " labels = {\"gini\" : \"Gini coefficient\", \"gdppc\" : \"GDP per capita\"}\n",
+ " )\n",
+ "fig.update_traces(textposition=\"bottom right\")\n",
+ "fig.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fff9bc38",
+ "metadata": {},
+ "source": [
+ "This plot shows that all three Western economies’ GDP per capita has grown over\n",
+ "time with some fluctuations in the Gini coefficient.\n",
+ "\n",
+ "From the early 80’s the United Kingdom and the US economies both saw increases\n",
+ "in income inequality.\n",
+ "\n",
+ "Interestingly, since the year 2000, the United Kingdom saw a decline in income inequality while\n",
+ "the US exhibits persistent but stable levels around a Gini coefficient of 40."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "37af6b9f",
+ "metadata": {},
+ "source": [
+ "## Top shares\n",
+ "\n",
+ "Another popular measure of inequality is the top shares.\n",
+ "\n",
+ "In this section we show how to compute top shares."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a0973931",
+ "metadata": {},
+ "source": [
+ "### Definition\n",
+ "\n",
+ "As before, suppose that the sample $ w_1, \\ldots, w_n $ has been sorted from smallest to largest.\n",
+ "\n",
+ "Given the Lorenz curve $ y = L(x) $ defined above, the top $ 100 \\times p \\% $\n",
+ "share is defined as"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f3a2b59e",
+ "metadata": {},
+ "source": [
+ "### \n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "T(p) = 1 - L (1-p) \n",
+ " \\approx \\frac{\\sum_{j\\geq i} w_j}{ \\sum_{j \\leq n} w_j}, \\quad i = \\lfloor n (1-p)\\rfloor \\tag{6.1}\n",
+ "$$\n",
+ "\n",
+ "Here $ \\lfloor \\cdot \\rfloor $ is the floor function, which rounds any\n",
+ "number down to the integer less than or equal to that number.\n",
+ "\n",
+ "The following code uses the data from dataframe `df_income_wealth` to generate another dataframe `df_topshares`.\n",
+ "\n",
+ "`df_topshares` stores the top 10 percent shares for the total income, the labor income and net wealth from 1950 to 2016 in US."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d34071dd",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# transfer the survey weights from absolute into relative values\n",
+ "df1 = df_income_wealth\n",
+ "df2 = df1.groupby('year').sum(numeric_only=True).reset_index()\n",
+ "df3 = df2[['year', 'weights']]\n",
+ "df3.columns = 'year', 'r_weights'\n",
+ "df4 = pd.merge(df3, df1, how=\"left\", on=[\"year\"])\n",
+ "df4['r_weights'] = df4['weights'] / df4['r_weights']\n",
+ "\n",
+ "# create weighted nw, ti, li\n",
+ "df4['weighted_n_wealth'] = df4['n_wealth'] * df4['r_weights']\n",
+ "df4['weighted_t_income'] = df4['t_income'] * df4['r_weights']\n",
+ "df4['weighted_l_income'] = df4['l_income'] * df4['r_weights']\n",
+ "\n",
+ "# extract two top 10% groups by net wealth and total income.\n",
+ "df6 = df4[df4['nw_groups'] == 'Top 10%']\n",
+ "df7 = df4[df4['ti_groups'] == 'Top 10%']\n",
+ "\n",
+ "# calculate the sum of weighted top 10% by net wealth,\n",
+ "# total income and labor income.\n",
+ "df5 = df4.groupby('year').sum(numeric_only=True).reset_index()\n",
+ "df8 = df6.groupby('year').sum(numeric_only=True).reset_index()\n",
+ "df9 = df7.groupby('year').sum(numeric_only=True).reset_index()\n",
+ "\n",
+ "df5['weighted_n_wealth_top10'] = df8['weighted_n_wealth']\n",
+ "df5['weighted_t_income_top10'] = df9['weighted_t_income']\n",
+ "df5['weighted_l_income_top10'] = df9['weighted_l_income']\n",
+ "\n",
+ "# calculate the top 10% shares of the three variables.\n",
+ "df5['topshare_n_wealth'] = df5['weighted_n_wealth_top10'] / \\\n",
+ " df5['weighted_n_wealth']\n",
+ "df5['topshare_t_income'] = df5['weighted_t_income_top10'] / \\\n",
+ " df5['weighted_t_income']\n",
+ "df5['topshare_l_income'] = df5['weighted_l_income_top10'] / \\\n",
+ " df5['weighted_l_income']\n",
+ "\n",
+ "# we only need these vars for top 10 percent shares\n",
+ "df_topshares = df5[['year', 'topshare_n_wealth',\n",
+ " 'topshare_t_income', 'topshare_l_income']]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "62e117a5",
+ "metadata": {},
+ "source": [
+ "Then let’s plot the top shares."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c81c03af",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax.plot(years, df_topshares[\"topshare_l_income\"],\n",
+ " marker='o', label=\"labor income\")\n",
+ "ax.plot(years, df_topshares[\"topshare_n_wealth\"],\n",
+ " marker='o', label=\"net wealth\")\n",
+ "ax.plot(years, df_topshares[\"topshare_t_income\"],\n",
+ " marker='o', label=\"total income\")\n",
+ "ax.set_xlabel(\"year\")\n",
+ "ax.set_ylabel(r\"top $10\\%$ share\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "26052ca1",
+ "metadata": {},
+ "source": [
+ "## Exercises"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d0c3174c",
+ "metadata": {},
+ "source": [
+ "## Exercise 6.1\n",
+ "\n",
+ "Using simulation, compute the top 10 percent shares for the collection of\n",
+ "lognormal distributions associated with the random variables $ w_\\sigma =\n",
+ "\\exp(\\mu + \\sigma Z) $, where $ Z \\sim N(0, 1) $ and $ \\sigma $ varies over a\n",
+ "finite grid between $ 0.2 $ and $ 4 $.\n",
+ "\n",
+ "As $ \\sigma $ increases, so does the variance of $ w_\\sigma $.\n",
+ "\n",
+ "To focus on volatility, adjust $ \\mu $ at each step to maintain the equality\n",
+ "$ \\mu=-\\sigma^2/2 $.\n",
+ "\n",
+ "For each $ \\sigma $, generate 2,000 independent draws of $ w_\\sigma $ and\n",
+ "calculate the Lorenz curve and Gini coefficient.\n",
+ "\n",
+ "Confirm that higher variance\n",
+ "generates more dispersion in the sample, and hence greater inequality."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b51ccfd5",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 6.1](https://intro.quantecon.org/#inequality_ex1)\n",
+ "\n",
+ "Here is one solution:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "560acc51",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def calculate_top_share(s, p=0.1):\n",
+ " \n",
+ " s = np.sort(s)\n",
+ " n = len(s)\n",
+ " index = int(n * (1 - p))\n",
+ " return s[index:].sum() / s.sum()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c06c67a0",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "k = 5\n",
+ "σ_vals = np.linspace(0.2, 4, k)\n",
+ "n = 2_000\n",
+ "\n",
+ "topshares = []\n",
+ "ginis = []\n",
+ "f_vals = []\n",
+ "l_vals = []\n",
+ "\n",
+ "for σ in σ_vals:\n",
+ " μ = -σ ** 2 / 2\n",
+ " y = np.exp(μ + σ * np.random.randn(n))\n",
+ " f_val, l_val = lorenz_curve(y)\n",
+ " f_vals.append(f_val)\n",
+ " l_vals.append(l_val)\n",
+ " ginis.append(gini_coefficient(y))\n",
+ " topshares.append(calculate_top_share(y))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5d1a644a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plot_inequality_measures(σ_vals, \n",
+ " topshares, \n",
+ " \"simulated data\", \n",
+ " \"$\\sigma$\", \n",
+ " \"top $10\\%$ share\") \n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c678f330",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plot_inequality_measures(σ_vals, \n",
+ " ginis, \n",
+ " \"simulated data\", \n",
+ " \"$\\sigma$\", \n",
+ " \"gini coefficient\")\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4ac15e6b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax.plot([0,1],[0,1], label=f\"equality\")\n",
+ "for i in range(len(f_vals)):\n",
+ " ax.plot(f_vals[i], l_vals[i], label=f\"$\\sigma$ = {σ_vals[i]}\")\n",
+ "plt.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1fdf8875",
+ "metadata": {},
+ "source": [
+ "## Exercise 6.2\n",
+ "\n",
+ "According to the definition of the top shares [(6.1)](#equation-topshares) we can also calculate the top percentile shares using the Lorenz curve.\n",
+ "\n",
+ "Compute the top shares of US net wealth using the corresponding Lorenz curves data: `f_vals_nw, l_vals_nw` and linear interpolation.\n",
+ "\n",
+ "Plot the top shares generated from Lorenz curve and the top shares approximated from data together."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fd5eae89",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 6.2](https://intro.quantecon.org/#inequality_ex2)\n",
+ "\n",
+ "Here is one solution:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "07b3273a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def lorenz2top(f_val, l_val, p=0.1):\n",
+ " t = lambda x: np.interp(x, f_val, l_val)\n",
+ " return 1- t(1 - p)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "765da26c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "top_shares_nw = []\n",
+ "for f_val, l_val in zip(f_vals_nw, l_vals_nw):\n",
+ " top_shares_nw.append(lorenz2top(f_val, l_val))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b3a678cc",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "\n",
+ "ax.plot(years, df_topshares[\"topshare_n_wealth\"], marker='o',\\\n",
+ " label=\"net wealth-approx\")\n",
+ "ax.plot(years, top_shares_nw, marker='o', label=\"net wealth-lorenz\")\n",
+ "\n",
+ "ax.set_xlabel(\"year\")\n",
+ "ax.set_ylabel(\"top $10\\%$ share\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cafc1117",
+ "metadata": {},
+ "source": [
+ "## Exercise 6.3\n",
+ "\n",
+ "The [code to compute the Gini coefficient is listed in the lecture above](#code-gini-coefficient).\n",
+ "\n",
+ "This code uses loops to calculate the coefficient based on income or wealth data.\n",
+ "\n",
+ "This function can be re-written using vectorization which will greatly improve the computational efficiency when using `python`.\n",
+ "\n",
+ "Re-write the function `gini_coefficient` using `numpy` and vectorized code.\n",
+ "\n",
+ "You can compare the output of this new function with the one above, and note the speed differences."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "73809e5a",
+ "metadata": {},
+ "source": [
+ "## Solution to[ Exercise 6.3](https://intro.quantecon.org/#inequality_ex3)\n",
+ "\n",
+ "Let’s take a look at some raw data for the US that is stored in `df_income_wealth`"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "16d20bff",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_income_wealth.describe()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b7c7242a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_income_wealth.head(n=4)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1c37eb66",
+ "metadata": {},
+ "source": [
+ "We will focus on wealth variable `n_wealth` to compute a Gini coefficient for the year 2016."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "87ec8694",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data = df_income_wealth[df_income_wealth.year == 2016].sample(3000, random_state=1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "23762d1a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "data.head(n=2)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b008f8cb",
+ "metadata": {},
+ "source": [
+ "We can first compute the Gini coefficient using the function defined in the lecture above."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "09952be5",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "gini_coefficient(data.n_wealth.values)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f0c9fd68",
+ "metadata": {},
+ "source": [
+ "Now we can write a vectorized version using `numpy`"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "0c130379",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def gini(y):\n",
+ " n = len(y)\n",
+ " y_1 = np.reshape(y, (n, 1))\n",
+ " y_2 = np.reshape(y, (1, n))\n",
+ " g_sum = np.sum(np.abs(y_1 - y_2))\n",
+ " return g_sum / (2 * n * np.sum(y))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "289dec8c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "gini(data.n_wealth.values)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "24db8d6c",
+ "metadata": {},
+ "source": [
+ "Let’s simulate five populations by drawing from a lognormal distribution as before"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b4e6ea08",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "k = 5\n",
+ "σ_vals = np.linspace(0.2, 4, k)\n",
+ "n = 2_000\n",
+ "σ_vals = σ_vals.reshape((k,1))\n",
+ "μ_vals = -σ_vals**2/2\n",
+ "y_vals = np.exp(μ_vals + σ_vals*np.random.randn(n))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9213a786",
+ "metadata": {},
+ "source": [
+ "We can compute the Gini coefficient for these five populations using the vectorized function, the computation time is shown below:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "82045aca",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "%%time\n",
+ "gini_coefficients =[]\n",
+ "for i in range(k):\n",
+ " gini_coefficients.append(gini(y_vals[i]))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7fb212d7",
+ "metadata": {},
+ "source": [
+ "This shows the vectorized function is much faster.\n",
+ "This gives us the Gini coefficients for these five households."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fe977341",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "gini_coefficients"
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476281.3215766,
+ "filename": "inequality.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Income and Wealth Inequality"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/inflation_history.ipynb b/_notebooks/inflation_history.ipynb
new file mode 100644
index 000000000..45446328f
--- /dev/null
+++ b/_notebooks/inflation_history.ipynb
@@ -0,0 +1,863 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "d88b8d6d",
+ "metadata": {},
+ "source": [
+ "# Price Level Histories\n",
+ "\n",
+ "This lecture offers some historical evidence about fluctuations in levels of aggregate price indexes.\n",
+ "\n",
+ "Let’s start by installing the necessary Python packages.\n",
+ "\n",
+ "The `xlrd` package is used by `pandas` to perform operations on Excel files."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d204be16",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "!pip install xlrd"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "02e79eb9",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "from importlib.metadata import version\n",
+ "from packaging.version import Version\n",
+ "\n",
+ "if Version(version(\"pandas\")) < Version('2.1.4'):\n",
+ " !pip install \"pandas>=2.1.4\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8d7ccfd2",
+ "metadata": {},
+ "source": [
+ "We can then import the Python modules we will use."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "55f88652",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt\n",
+ "import matplotlib.dates as mdates"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c4bcb954",
+ "metadata": {},
+ "source": [
+ "The rate of growth of the price level is called **inflation** in the popular press and in discussions among central bankers and treasury officials.\n",
+ "\n",
+ "The price level is measured in units of domestic currency per units of a representative bundle of consumption goods.\n",
+ "\n",
+ "Thus, in the US, the price level at $ t $ is measured in dollars (month $ t $ or year $ t $) per unit of the consumption bundle.\n",
+ "\n",
+ "Until the early 20th century, in many western economies, price levels fluctuated from year to year but didn’t have much of a trend.\n",
+ "\n",
+ "Often the price levels ended a century near where they started.\n",
+ "\n",
+ "Things were different in the 20th century, as we shall see in this lecture.\n",
+ "\n",
+ "A widely believed explanation of this big difference is that countries’ abandoning gold and silver standards in the early twentieth century.\n",
+ "\n",
+ "This lecture sets the stage for some subsequent lectures about a theory that macro economists use to think about determinants of the price level, namely, [A Monetarist Theory of Price Levels](https://intro.quantecon.org/cagan_ree.html) and [Monetarist Theory of Price Levels with Adaptive Expectations](https://intro.quantecon.org/cagan_adaptive.html)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3acc01a8",
+ "metadata": {},
+ "source": [
+ "## Four centuries of price levels\n",
+ "\n",
+ "We begin by displaying data that originally appeared on page 35 of [[Sargent and Velde, 2002](https://intro.quantecon.org/zreferences.html#id12)] that show price levels for four “hard currency” countries from 1600 to 1914.\n",
+ "\n",
+ "- France \n",
+ "- Spain (Castile) \n",
+ "- United Kingdom \n",
+ "- United States \n",
+ "\n",
+ "\n",
+ "In the present context, the phrase “hard currency” means that the countries were on a commodity-money standard: money consisted of gold and silver coins that circulated at values largely determined by the weights of their gold and silver contents.\n",
+ "\n",
+ ">**Note**\n",
+ ">\n",
+ ">Under a gold or silver standard, some money also consisted of “warehouse certificates” that represented paper claims on gold or silver coins. Bank notes issued by the government or private banks can be viewed as examples of such “warehouse certificates”.\n",
+ "\n",
+ "Let us bring the data into pandas from a spreadsheet that is [hosted on github](https://github.com/QuantEcon/lecture-python-intro/tree/main/lectures/datasets)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a5f69467",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Import data and clean up the index\n",
+ "data_url = \"https://github.com/QuantEcon/lecture-python-intro/raw/main/lectures/datasets/longprices.xls\"\n",
+ "df_fig5 = pd.read_excel(data_url, \n",
+ " sheet_name='all', \n",
+ " header=2, \n",
+ " index_col=0).iloc[1:]\n",
+ "df_fig5.index = df_fig5.index.astype(int)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9ab9c8e6",
+ "metadata": {},
+ "source": [
+ "We first plot price levels over the period 1600-1914.\n",
+ "\n",
+ "During most years in this time interval, the countries were on a gold or silver standard."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "253d992a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "df_fig5_befe1914 = df_fig5[df_fig5.index <= 1914]\n",
+ "\n",
+ "# Create plot\n",
+ "cols = ['UK', 'US', 'France', 'Castile']\n",
+ "\n",
+ "fig, ax = plt.subplots(figsize=(10,6))\n",
+ "\n",
+ "for col in cols:\n",
+ " ax.plot(df_fig5_befe1914.index, \n",
+ " df_fig5_befe1914[col], label=col, lw=2)\n",
+ "\n",
+ "ax.legend()\n",
+ "ax.set_ylabel('Index 1913 = 100')\n",
+ "ax.set_xlabel('Year')\n",
+ "ax.set_xlim(xmin=1600)\n",
+ "plt.tight_layout()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4f75ec41",
+ "metadata": {},
+ "source": [
+ "We say “most years” because there were temporary lapses from the gold or silver standard.\n",
+ "\n",
+ "By staring at Fig. 4.1 carefully, you might be able to guess when these temporary lapses occurred, because they were also times during which price levels temporarily rose markedly:\n",
+ "\n",
+ "- 1791-1797 in France (French Revolution) \n",
+ "- 1776-1790 in the US (War for Independence from Great Britain) \n",
+ "- 1861-1865 in the US (Civil War) \n",
+ "\n",
+ "\n",
+ "During these episodes, the gold/silver standard was temporarily abandoned when a government printed paper money to pay for war expenditures.\n",
+ "\n",
+ ">**Note**\n",
+ ">\n",
+ ">This quantecon lecture [Inflation During French Revolution](https://intro.quantecon.org/french_rev.html) describes circumstances leading up to and during the big inflation that occurred during the French Revolution.\n",
+ "\n",
+ "Despite these temporary lapses, a striking thing about the figure is that price levels were roughly constant over three centuries.\n",
+ "\n",
+ "In the early century, two other features of this data attracted the attention of [Irving Fisher](https://en.wikipedia.org/wiki/Irving_Fisher) of Yale University and [John Maynard Keynes](https://en.wikipedia.org/wiki/John_Maynard_Keynes) of Cambridge University.\n",
+ "\n",
+ "- Despite being anchored to the same average level over long time spans, there were considerable year-to-year variations in price levels \n",
+ "- While using valuable gold and silver as coins succeeded in anchoring the price level by limiting the supply of money, it cost real resources. \n",
+ "- a country paid a high “opportunity cost” for using gold and silver coins as money – that gold and silver could instead have been made into valuable jewelry and other durable goods. \n",
+ "\n",
+ "\n",
+ "Keynes and Fisher proposed what they claimed would be a more efficient way to achieve a price level that\n",
+ "\n",
+ "- would be at least as firmly anchored as achieved under a gold or silver standard, and \n",
+ "- would also exhibit less year-to-year short-term fluctuations. \n",
+ "\n",
+ "\n",
+ "They said that central bank could achieve price level stability by\n",
+ "\n",
+ "- issuing **limited supplies** of paper currency \n",
+ "- refusing to print money to finance government expenditures \n",
+ "\n",
+ "\n",
+ "This logic prompted John Maynard Keynes to call a commodity standard a “barbarous relic.”\n",
+ "\n",
+ "A paper currency or “fiat money” system disposes of all reserves behind a currency.\n",
+ "\n",
+ "But adhering to a gold or silver standard had provided an automatic mechanism for limiting the supply of money, thereby anchoring the price level.\n",
+ "\n",
+ "To anchor the price level, a pure paper or fiat money system replaces that automatic mechanism with a central bank with the authority and determination to limit the supply of money (and to deter counterfeiters!)\n",
+ "\n",
+ "Now let’s see what happened to the price level in the four countries after 1914, when one after another of them left the gold/silver standard by showing the complete graph that originally appeared on page 35 of [[Sargent and Velde, 2002](https://intro.quantecon.org/zreferences.html#id12)].\n",
+ "\n",
+ "Fig. 4.2 shows the logarithm of price levels over four “hard currency” countries from 1600 to 2000.\n",
+ "\n",
+ ">**Note**\n",
+ ">\n",
+ ">Although we didn’t have to use logarithms in our earlier graphs that had stopped in 1914, we now choose to use logarithms because we want to fit observations after 1914 in the same graph as the earlier observations.\n",
+ "\n",
+ "After the outbreak of the Great War in 1914, the four countries left the gold standard and in so doing acquired the ability to print money to finance government expenditures."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c2554157",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots(dpi=200)\n",
+ "\n",
+ "for col in cols:\n",
+ " ax.plot(df_fig5.index, df_fig5[col], lw=2)\n",
+ " ax.text(x=df_fig5.index[-1]+2, \n",
+ " y=df_fig5[col].iloc[-1], s=col)\n",
+ "\n",
+ "ax.set_yscale('log')\n",
+ "ax.set_ylabel('Logs of price levels (Index 1913 = 100)')\n",
+ "ax.set_ylim([10, 1e6])\n",
+ "ax.set_xlabel('year')\n",
+ "ax.set_xlim(xmin=1600)\n",
+ "plt.tight_layout()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1983e7ef",
+ "metadata": {},
+ "source": [
+ "Fig. 4.2 shows that paper-money-printing central banks didn’t do as well as the gold and standard silver standard in anchoring price levels.\n",
+ "\n",
+ "That would probably have surprised or disappointed Irving Fisher and John Maynard Keynes.\n",
+ "\n",
+ "Actually, earlier economists and statesmen knew about the possibility of fiat money systems long before Keynes and Fisher advocated them in the early 20th century.\n",
+ "\n",
+ "Proponents of a commodity money system did not trust governments and central banks properly to manage a fiat money system.\n",
+ "\n",
+ "They were willing to pay the resource costs associated with setting up and maintaining a commodity money system.\n",
+ "\n",
+ "In light of the high and persistent inflation that many countries experienced after they abandoned commodity monies in the twentieth century, we hesitate to criticize advocates of a gold or silver standard for their preference to stay on the pre-1914 gold/silver standard.\n",
+ "\n",
+ "The breadth and lengths of the inflationary experiences of the twentieth century under paper money fiat standards are historically unprecedented."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "92a31293",
+ "metadata": {},
+ "source": [
+ "## Four big inflations\n",
+ "\n",
+ "In the wake of World War I, which ended in November 1918, monetary and fiscal authorities struggled to achieve price level stability without being on a gold or silver standard.\n",
+ "\n",
+ "We present four graphs from “The Ends of Four Big Inflations” from chapter 3 of [[Sargent, 2013](https://intro.quantecon.org/zreferences.html#id13)].\n",
+ "\n",
+ "The graphs depict logarithms of price levels during the early post World War I years for four countries:\n",
+ "\n",
+ "- Figure 3.1, Retail prices Austria, 1921-1924 (page 42) \n",
+ "- Figure 3.2, Wholesale prices Hungary, 1921-1924 (page 43) \n",
+ "- Figure 3.3, Wholesale prices, Poland, 1921-1924 (page 44) \n",
+ "- Figure 3.4, Wholesale prices, Germany, 1919-1924 (page 45) \n",
+ "\n",
+ "\n",
+ "We have added logarithms of the exchange rates vis-à-vis the US dollar to each of the four graphs\n",
+ "from chapter 3 of [[Sargent, 2013](https://intro.quantecon.org/zreferences.html#id13)].\n",
+ "\n",
+ "Data underlying our graphs appear in tables in an appendix to chapter 3 of [[Sargent, 2013](https://intro.quantecon.org/zreferences.html#id13)].\n",
+ "We have transcribed all of these data into a spreadsheet chapter_3.xlsx that we read into pandas.\n",
+ "\n",
+ "In the code cell below we clean the data and build a `pandas.dataframe`."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9e23f9ad",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def process_entry(entry):\n",
+ " \"Clean each entry of a dataframe.\"\n",
+ " \n",
+ " if type(entry) == str:\n",
+ " # Remove leading and trailing whitespace\n",
+ " entry = entry.strip()\n",
+ " # Remove comma\n",
+ " entry = entry.replace(',', '')\n",
+ " \n",
+ " # Remove HTML markers\n",
+ " item_to_remove = ['\n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c7f769bb",
+ "metadata": {},
+ "source": [
+ "# Univariate Time Series with Matrix Algebra"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "846a7275",
+ "metadata": {},
+ "source": [
+ "## Overview\n",
+ "\n",
+ "This lecture uses matrices to solve some linear difference equations.\n",
+ "\n",
+ "As a running example, we’ll study a **second-order linear difference\n",
+ "equation** that was the key technical tool in Paul Samuelson’s 1939\n",
+ "article [[Samuelson, 1939](https://intro.quantecon.org/zreferences.html#id107)] that introduced the *multiplier-accelerator model*.\n",
+ "\n",
+ "This model became the workhorse that powered early econometric versions of\n",
+ "Keynesian macroeconomic models in the United States.\n",
+ "\n",
+ "You can read about the details of that model in [Samuelson Multiplier-Accelerator](https://python.quantecon.org/samuelson.html).\n",
+ "\n",
+ "(That lecture also describes some technicalities about second-order linear difference equations.)\n",
+ "\n",
+ "In this lecture, we’ll also learn about an **autoregressive** representation and a **moving average** representation of a non-stationary\n",
+ "univariate time series $ \\{y_t\\}_{t=0}^T $.\n",
+ "\n",
+ "We’ll also study a “perfect foresight” model of stock prices that involves solving\n",
+ "a “forward-looking” linear difference equation.\n",
+ "\n",
+ "We will use the following imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ad044a0f",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from matplotlib import cm\n",
+ "\n",
+ "# Custom figsize for this lecture\n",
+ "plt.rcParams[\"figure.figsize\"] = (11, 5)\n",
+ "\n",
+ "# Set decimal printing to 3 decimal places\n",
+ "np.set_printoptions(precision=3, suppress=True)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c00a9d26",
+ "metadata": {},
+ "source": [
+ "## Samuelson’s model\n",
+ "\n",
+ "Let $ t = 0, \\pm 1, \\pm 2, \\ldots $ index time.\n",
+ "\n",
+ "For $ t = 1, 2, 3, \\ldots, T $ suppose that\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_{t} = \\alpha_{0} + \\alpha_{1} y_{t-1} + \\alpha_{2} y_{t-2} \\tag{36.1}\n",
+ "$$\n",
+ "\n",
+ "where we assume that $ y_0 $ and $ y_{-1} $ are given numbers\n",
+ "that we take as *initial conditions*.\n",
+ "\n",
+ "In Samuelson’s model, $ y_t $ stood for **national income** or perhaps a different\n",
+ "measure of aggregate activity called **gross domestic product** (GDP) at time $ t $.\n",
+ "\n",
+ "Equation [(36.1)](#equation-tswm-1) is called a *second-order linear difference equation*. It is called second order because it depends on two lags.\n",
+ "\n",
+ "But actually, it is a collection of $ T $ simultaneous linear\n",
+ "equations in the $ T $ variables $ y_1, y_2, \\ldots, y_T $.\n",
+ "\n",
+ ">**Note**\n",
+ ">\n",
+ ">To be able to solve a second-order linear difference\n",
+ "equation, we require two *boundary conditions* that can take the form\n",
+ "either of two *initial conditions*, two *terminal conditions* or\n",
+ "possibly one of each.\n",
+ "\n",
+ "Let’s write our equations as a stacked system\n",
+ "\n",
+ "$$\n",
+ "\\underset{\\equiv A}{\\underbrace{\\left[\\begin{array}{cccccccc}\n",
+ "1 & 0 & 0 & 0 & \\cdots & 0 & 0 & 0\\\\\n",
+ "-\\alpha_{1} & 1 & 0 & 0 & \\cdots & 0 & 0 & 0\\\\\n",
+ "-\\alpha_{2} & -\\alpha_{1} & 1 & 0 & \\cdots & 0 & 0 & 0\\\\\n",
+ "0 & -\\alpha_{2} & -\\alpha_{1} & 1 & \\cdots & 0 & 0 & 0\\\\\n",
+ "\\vdots & \\vdots & \\vdots & \\vdots & \\cdots & \\vdots & \\vdots & \\vdots\\\\\n",
+ "0 & 0 & 0 & 0 & \\cdots & -\\alpha_{2} & -\\alpha_{1} & 1\n",
+ "\\end{array}\\right]}}\\left[\\begin{array}{c}\n",
+ "y_{1}\\\\\n",
+ "y_{2}\\\\\n",
+ "y_{3}\\\\\n",
+ "y_{4}\\\\\n",
+ "\\vdots\\\\\n",
+ "y_{T}\n",
+ "\\end{array}\\right]=\\underset{\\equiv b}{\\underbrace{\\left[\\begin{array}{c}\n",
+ "\\alpha_{0}+\\alpha_{1}y_{0}+\\alpha_{2}y_{-1}\\\\\n",
+ "\\alpha_{0}+\\alpha_{2}y_{0}\\\\\n",
+ "\\alpha_{0}\\\\\n",
+ "\\alpha_{0}\\\\\n",
+ "\\vdots\\\\\n",
+ "\\alpha_{0}\n",
+ "\\end{array}\\right]}}\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "A y = b\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "$$\n",
+ "y = \\begin{bmatrix} y_1 \\cr y_2 \\cr \\vdots \\cr y_T \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "Evidently $ y $ can be computed from\n",
+ "\n",
+ "$$\n",
+ "y = A^{-1} b\n",
+ "$$\n",
+ "\n",
+ "The vector $ y $ is a complete time path $ \\{y_t\\}_{t=1}^T $.\n",
+ "\n",
+ "Let’s put Python to work on an example that captures the flavor of\n",
+ "Samuelson’s multiplier-accelerator model.\n",
+ "\n",
+ "We’ll set parameters equal to the same values we used in [Samuelson Multiplier-Accelerator](https://python.quantecon.org/samuelson.html)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e3f57875",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "T = 80\n",
+ "\n",
+ "# parameters\n",
+ "α_0 = 10.0\n",
+ "α_1 = 1.53\n",
+ "α_2 = -.9\n",
+ "\n",
+ "y_neg1 = 28.0 # y_{-1}\n",
+ "y_0 = 24.0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "acb16430",
+ "metadata": {},
+ "source": [
+ "Now we construct $ A $ and $ b $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e98934d0",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "A = np.identity(T) # The T x T identity matrix\n",
+ "\n",
+ "for i in range(T):\n",
+ "\n",
+ " if i-1 >= 0:\n",
+ " A[i, i-1] = -α_1\n",
+ "\n",
+ " if i-2 >= 0:\n",
+ " A[i, i-2] = -α_2\n",
+ "\n",
+ "b = np.full(T, α_0)\n",
+ "b[0] = α_0 + α_1 * y_0 + α_2 * y_neg1\n",
+ "b[1] = α_0 + α_2 * y_0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "10caa21b",
+ "metadata": {},
+ "source": [
+ "Let’s look at the matrix $ A $ and the vector $ b $ for our\n",
+ "example."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5bbe22f4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "A, b"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a7248299",
+ "metadata": {},
+ "source": [
+ "Now let’s solve for the path of $ y $.\n",
+ "\n",
+ "If $ y_t $ is GNP at time $ t $, then we have a version of\n",
+ "Samuelson’s model of the dynamics for GNP.\n",
+ "\n",
+ "To solve $ y = A^{-1} b $ we can either invert $ A $ directly, as in"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "86b75202",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "A_inv = np.linalg.inv(A)\n",
+ "\n",
+ "y = A_inv @ b"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "aaa1512c",
+ "metadata": {},
+ "source": [
+ "or we can use `np.linalg.solve`:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "dd358100",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "y_second_method = np.linalg.solve(A, b)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "48a96d0b",
+ "metadata": {},
+ "source": [
+ "Here make sure the two methods give the same result, at least up to floating\n",
+ "point precision:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f57c4933",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "np.allclose(y, y_second_method)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5c73514a",
+ "metadata": {},
+ "source": [
+ "$ A $ is invertible as it is lower triangular and [its diagonal entries are non-zero](https://www.statlect.com/matrix-algebra/triangular-matrix)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "185c1ec5",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Check if A is lower triangular\n",
+ "np.allclose(A, np.tril(A))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0cc209a4",
+ "metadata": {},
+ "source": [
+ ">**Note**\n",
+ ">\n",
+ ">In general, `np.linalg.solve` is more numerically stable than using\n",
+ "`np.linalg.inv` directly.\n",
+ "However, stability is not an issue for this small example. Moreover, we will\n",
+ "repeatedly use `A_inv` in what follows, so there is added value in computing\n",
+ "it directly.\n",
+ "\n",
+ "Now we can plot."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "21da560d",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plt.plot(np.arange(T)+1, y)\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6d7c43a1",
+ "metadata": {},
+ "source": [
+ "The [*steady state*](https://intro.quantecon.org/scalar_dynam.html#scalar-dynam-steady-state) value $ y^* $ of $ y_t $ is obtained by setting $ y_t = y_{t-1} =\n",
+ "y_{t-2} = y^* $ in [(36.1)](#equation-tswm-1), which yields\n",
+ "\n",
+ "$$\n",
+ "y^* = \\frac{\\alpha_{0}}{1 - \\alpha_{1} - \\alpha_{2}}\n",
+ "$$\n",
+ "\n",
+ "If we set the initial values to $ y_{0} = y_{-1} = y^* $, then $ y_{t} $ will be\n",
+ "constant:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d38fc0b9",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "y_star = α_0 / (1 - α_1 - α_2)\n",
+ "y_neg1_steady = y_star # y_{-1}\n",
+ "y_0_steady = y_star\n",
+ "\n",
+ "b_steady = np.full(T, α_0)\n",
+ "b_steady[0] = α_0 + α_1 * y_0_steady + α_2 * y_neg1_steady\n",
+ "b_steady[1] = α_0 + α_2 * y_0_steady"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c3acd850",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "y_steady = A_inv @ b_steady"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1ccdc7a7",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plt.plot(np.arange(T)+1, y_steady)\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3f12b4db",
+ "metadata": {},
+ "source": [
+ "## Adding a random term\n",
+ "\n",
+ "To generate some excitement, we’ll follow in the spirit of the great economists\n",
+ "[Eugen Slutsky](https://en.wikipedia.org/wiki/Eugen_Slutsky) and [Ragnar Frisch](https://en.wikipedia.org/wiki/Ragnar_Frisch) and replace our original second-order difference\n",
+ "equation with the following **second-order stochastic linear difference\n",
+ "equation**:\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y_{t} = \\alpha_{0} + \\alpha_{1} y_{t-1} + \\alpha_{2} y_{t-2} + u_t \\tag{36.2}\n",
+ "$$\n",
+ "\n",
+ "where $ u_{t} \\sim N\\left(0, \\sigma_{u}^{2}\\right) $ and is [IID](https://intro.quantecon.org/lln_clt.html#iid-theorem),\n",
+ "meaning independent and identically distributed.\n",
+ "\n",
+ "We’ll stack these $ T $ equations into a system cast in terms of\n",
+ "matrix algebra.\n",
+ "\n",
+ "Let’s define the random vector\n",
+ "\n",
+ "$$\n",
+ "u=\\left[\\begin{array}{c}\n",
+ "u_{1}\\\\\n",
+ "u_{2}\\\\\n",
+ "\\vdots\\\\\n",
+ "u_{T}\n",
+ "\\end{array}\\right]\n",
+ "$$\n",
+ "\n",
+ "Where $ A, b, y $ are defined as above, now assume that $ y $ is\n",
+ "governed by the system\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "A y = b + u \\tag{36.3}\n",
+ "$$\n",
+ "\n",
+ "The solution for $ y $ becomes\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "y = A^{-1} \\left(b + u\\right) \\tag{36.4}\n",
+ "$$\n",
+ "\n",
+ "Let’s try it out in Python."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "94fcc7de",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "σ_u = 2.\n",
+ "u = np.random.normal(0, σ_u, size=T)\n",
+ "y = A_inv @ (b + u)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "40fb8d40",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plt.plot(np.arange(T)+1, y)\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "59966e6e",
+ "metadata": {},
+ "source": [
+ "The above time series looks a lot like (detrended) GDP series for a\n",
+ "number of advanced countries in recent decades.\n",
+ "\n",
+ "We can simulate $ N $ paths."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "0ebf55d9",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "N = 100\n",
+ "\n",
+ "for i in range(N):\n",
+ " col = cm.viridis(np.random.rand()) # Choose a random color from viridis\n",
+ " u = np.random.normal(0, σ_u, size=T)\n",
+ " y = A_inv @ (b + u)\n",
+ " plt.plot(np.arange(T)+1, y, lw=0.5, color=col)\n",
+ "\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f916e9b3",
+ "metadata": {},
+ "source": [
+ "Also consider the case when $ y_{0} $ and $ y_{-1} $ are at\n",
+ "steady state."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "0828dd54",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "N = 100\n",
+ "\n",
+ "for i in range(N):\n",
+ " col = cm.viridis(np.random.rand()) # Choose a random color from viridis\n",
+ " u = np.random.normal(0, σ_u, size=T)\n",
+ " y_steady = A_inv @ (b_steady + u)\n",
+ " plt.plot(np.arange(T)+1, y_steady, lw=0.5, color=col)\n",
+ "\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f6400f14",
+ "metadata": {},
+ "source": [
+ "## Computing population moments\n",
+ "\n",
+ "We can apply standard formulas for multivariate normal distributions to compute the mean vector and covariance matrix\n",
+ "for our time series model\n",
+ "\n",
+ "$$\n",
+ "y = A^{-1} (b + u) .\n",
+ "$$\n",
+ "\n",
+ "You can read about multivariate normal distributions in this lecture [Multivariate Normal Distribution](https://python.quantecon.org/multivariate_normal.html).\n",
+ "\n",
+ "Let’s write our model as\n",
+ "\n",
+ "$$\n",
+ "y = \\tilde A (b + u)\n",
+ "$$\n",
+ "\n",
+ "where $ \\tilde A = A^{-1} $.\n",
+ "\n",
+ "Because linear combinations of normal random variables are normal, we know that\n",
+ "\n",
+ "$$\n",
+ "y \\sim {\\mathcal N}(\\mu_y, \\Sigma_y)\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "$$\n",
+ "\\mu_y = \\tilde A b\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "$$\n",
+ "\\Sigma_y = \\tilde A (\\sigma_u^2 I_{T \\times T} ) \\tilde A^T\n",
+ "$$\n",
+ "\n",
+ "Let’s write a Python class that computes the mean vector $ \\mu_y $ and covariance matrix $ \\Sigma_y $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "965d8def",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "class population_moments:\n",
+ " \"\"\"\n",
+ " Compute population moments μ_y, Σ_y.\n",
+ " ---------\n",
+ " Parameters:\n",
+ " α_0, α_1, α_2, T, y_neg1, y_0\n",
+ " \"\"\"\n",
+ " def __init__(self, α_0=10.0, \n",
+ " α_1=1.53, \n",
+ " α_2=-.9, \n",
+ " T=80, \n",
+ " y_neg1=28.0, \n",
+ " y_0=24.0, \n",
+ " σ_u=1):\n",
+ "\n",
+ " # compute A\n",
+ " A = np.identity(T)\n",
+ "\n",
+ " for i in range(T):\n",
+ " if i-1 >= 0:\n",
+ " A[i, i-1] = -α_1\n",
+ "\n",
+ " if i-2 >= 0:\n",
+ " A[i, i-2] = -α_2\n",
+ "\n",
+ " # compute b\n",
+ " b = np.full(T, α_0)\n",
+ " b[0] = α_0 + α_1 * y_0 + α_2 * y_neg1\n",
+ " b[1] = α_0 + α_2 * y_0\n",
+ "\n",
+ " # compute A inverse\n",
+ " A_inv = np.linalg.inv(A)\n",
+ "\n",
+ " self.A, self.b, self.A_inv, self.σ_u, self.T = A, b, A_inv, σ_u, T\n",
+ " \n",
+ " def sample_y(self, n):\n",
+ " \"\"\"\n",
+ " Give a sample of size n of y.\n",
+ " \"\"\"\n",
+ " A_inv, σ_u, b, T = self.A_inv, self.σ_u, self.b, self.T\n",
+ " us = np.random.normal(0, σ_u, size=[n, T])\n",
+ " ys = np.vstack([A_inv @ (b + u) for u in us])\n",
+ "\n",
+ " return ys\n",
+ "\n",
+ " def get_moments(self):\n",
+ " \"\"\"\n",
+ " Compute the population moments of y.\n",
+ " \"\"\"\n",
+ " A_inv, σ_u, b = self.A_inv, self.σ_u, self.b\n",
+ "\n",
+ " # compute μ_y\n",
+ " self.μ_y = A_inv @ b\n",
+ " self.Σ_y = σ_u**2 * (A_inv @ A_inv.T)\n",
+ " \n",
+ " return self.μ_y, self.Σ_y\n",
+ "\n",
+ "\n",
+ "series_process = population_moments()\n",
+ " \n",
+ "μ_y, Σ_y = series_process.get_moments()\n",
+ "A_inv = series_process.A_inv"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "54ff0b19",
+ "metadata": {},
+ "source": [
+ "It is enlightening to study the $ \\mu_y, \\Sigma_y $’s implied by various parameter values.\n",
+ "\n",
+ "Among other things, we can use the class to exhibit how **statistical stationarity** of $ y $ prevails only for very special initial conditions.\n",
+ "\n",
+ "Let’s begin by generating $ N $ time realizations of $ y $ plotting them together with population mean $ \\mu_y $ ."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b1a0419d",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Plot mean\n",
+ "N = 100\n",
+ "\n",
+ "for i in range(N):\n",
+ " col = cm.viridis(np.random.rand()) # Choose a random color from viridis\n",
+ " ys = series_process.sample_y(N)\n",
+ " plt.plot(ys[i,:], lw=0.5, color=col)\n",
+ " plt.plot(μ_y, color='red')\n",
+ "\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "21efd990",
+ "metadata": {},
+ "source": [
+ "Visually, notice how the variance across realizations of $ y_t $ decreases as $ t $ increases.\n",
+ "\n",
+ "Let’s plot the population variance of $ y_t $ against $ t $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "15e47b0d",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Plot variance\n",
+ "plt.plot(Σ_y.diagonal())\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "74d16dc5",
+ "metadata": {},
+ "source": [
+ "Notice how the population variance increases and asymptotes.\n",
+ "\n",
+ "Let’s print out the covariance matrix $ \\Sigma_y $ for a time series $ y $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "35370241",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "series_process = population_moments(α_0=0, \n",
+ " α_1=.8, \n",
+ " α_2=0, \n",
+ " T=6,\n",
+ " y_neg1=0., \n",
+ " y_0=0., \n",
+ " σ_u=1)\n",
+ "\n",
+ "μ_y, Σ_y = series_process.get_moments()\n",
+ "print(\"μ_y = \", μ_y)\n",
+ "print(\"Σ_y = \\n\", Σ_y)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9e489831",
+ "metadata": {},
+ "source": [
+ "Notice that the covariance between $ y_t $ and $ y_{t-1} $ – the elements on the superdiagonal – are *not* identical.\n",
+ "\n",
+ "This is an indication that the time series represented by our $ y $ vector is not **stationary**.\n",
+ "\n",
+ "To make it stationary, we’d have to alter our system so that our *initial conditions* $ (y_0, y_{-1}) $ are not fixed numbers but instead a jointly normally distributed random vector with a particular mean and covariance matrix.\n",
+ "\n",
+ "We describe how to do that in [Linear State Space Models](https://python.quantecon.org/linear_models.html).\n",
+ "\n",
+ "But just to set the stage for that analysis, let’s print out the bottom right corner of $ \\Sigma_y $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "dbb5e703",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "series_process = population_moments()\n",
+ "μ_y, Σ_y = series_process.get_moments()\n",
+ "\n",
+ "print(\"bottom right corner of Σ_y = \\n\", Σ_y[72:,72:])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b55545f1",
+ "metadata": {},
+ "source": [
+ "Please notice how the subdiagonal and superdiagonal elements seem to have converged.\n",
+ "\n",
+ "This is an indication that our process is asymptotically stationary.\n",
+ "\n",
+ "You can read about stationarity of more general linear time series models in this lecture [Linear State Space Models](https://python.quantecon.org/linear_models.html).\n",
+ "\n",
+ "There is a lot to be learned about the process by staring at the off diagonal elements of $ \\Sigma_y $ corresponding to different time periods $ t $, but we resist the temptation to do so here."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "dc6e18de",
+ "metadata": {},
+ "source": [
+ "## Moving average representation\n",
+ "\n",
+ "Let’s print out $ A^{-1} $ and stare at its structure\n",
+ "\n",
+ "- is it triangular or almost triangular or $ \\ldots $ ? \n",
+ "\n",
+ "\n",
+ "To study the structure of $ A^{-1} $, we shall print just up to $ 3 $ decimals.\n",
+ "\n",
+ "Let’s begin by printing out just the upper left hand corner of $ A^{-1} $."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ec27f4d4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "print(A_inv[0:7,0:7])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "595408ea",
+ "metadata": {},
+ "source": [
+ "Evidently, $ A^{-1} $ is a lower triangular matrix.\n",
+ "\n",
+ "Notice how every row ends with the previous row’s pre-diagonal entries.\n",
+ "\n",
+ "Since $ A^{-1} $ is lower triangular, each row represents $ y_t $ for a particular $ t $ as the sum of\n",
+ "\n",
+ "- a time-dependent function $ A^{-1} b $ of the initial conditions incorporated in $ b $, and \n",
+ "- a weighted sum of current and past values of the IID shocks $ \\{u_t\\} $. \n",
+ "\n",
+ "\n",
+ "Thus, let $ \\tilde{A}=A^{-1} $.\n",
+ "\n",
+ "Evidently, for $ t\\geq0 $,\n",
+ "\n",
+ "$$\n",
+ "y_{t+1}=\\sum_{i=1}^{t+1}\\tilde{A}_{t+1,i}b_{i}+\\sum_{i=1}^{t}\\tilde{A}_{t+1,i}u_{i}+u_{t+1}\n",
+ "$$\n",
+ "\n",
+ "This is a **moving average** representation with time-varying coefficients.\n",
+ "\n",
+ "Just as system [(36.4)](#equation-eq-eqma) constitutes a\n",
+ "**moving average** representation for $ y $, system [(36.3)](#equation-eq-eqar) constitutes an **autoregressive** representation for $ y $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "ab60f149",
+ "metadata": {},
+ "source": [
+ "## A forward looking model\n",
+ "\n",
+ "Samuelson’s model is *backward looking* in the sense that we give it *initial conditions* and let it\n",
+ "run.\n",
+ "\n",
+ "Let’s now turn to model that is *forward looking*.\n",
+ "\n",
+ "We apply similar linear algebra machinery to study a *perfect\n",
+ "foresight* model widely used as a benchmark in macroeconomics and\n",
+ "finance.\n",
+ "\n",
+ "As an example, we suppose that $ p_t $ is the price of a stock and\n",
+ "that $ y_t $ is its dividend.\n",
+ "\n",
+ "We assume that $ y_t $ is determined by second-order difference\n",
+ "equation that we analyzed just above, so that\n",
+ "\n",
+ "$$\n",
+ "y = A^{-1} \\left(b + u\\right)\n",
+ "$$\n",
+ "\n",
+ "Our *perfect foresight* model of stock prices is\n",
+ "\n",
+ "$$\n",
+ "p_{t} = \\sum_{j=0}^{T-t} \\beta^{j} y_{t+j}, \\quad \\beta \\in (0,1)\n",
+ "$$\n",
+ "\n",
+ "where $ \\beta $ is a discount factor.\n",
+ "\n",
+ "The model asserts that the price of the stock at $ t $ equals the\n",
+ "discounted present values of the (perfectly foreseen) future dividends.\n",
+ "\n",
+ "Form\n",
+ "\n",
+ "$$\n",
+ "\\underset{\\equiv p}{\\underbrace{\\left[\\begin{array}{c}\n",
+ "p_{1}\\\\\n",
+ "p_{2}\\\\\n",
+ "p_{3}\\\\\n",
+ "\\vdots\\\\\n",
+ "p_{T}\n",
+ "\\end{array}\\right]}}=\\underset{\\equiv B}{\\underbrace{\\left[\\begin{array}{ccccc}\n",
+ "1 & \\beta & \\beta^{2} & \\cdots & \\beta^{T-1}\\\\\n",
+ "0 & 1 & \\beta & \\cdots & \\beta^{T-2}\\\\\n",
+ "0 & 0 & 1 & \\cdots & \\beta^{T-3}\\\\\n",
+ "\\vdots & \\vdots & \\vdots & \\vdots & \\vdots\\\\\n",
+ "0 & 0 & 0 & \\cdots & 1\n",
+ "\\end{array}\\right]}}\\left[\\begin{array}{c}\n",
+ "y_{1}\\\\\n",
+ "y_{2}\\\\\n",
+ "y_{3}\\\\\n",
+ "\\vdots\\\\\n",
+ "y_{T}\n",
+ "\\end{array}\\right]\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d4efa916",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "β = .96"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4a9adba4",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# construct B\n",
+ "B = np.zeros((T, T))\n",
+ "\n",
+ "for i in range(T):\n",
+ " B[i, i:] = β ** np.arange(0, T-i)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "44a0b028",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "print(B)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "43664092",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "σ_u = 0.\n",
+ "u = np.random.normal(0, σ_u, size=T)\n",
+ "y = A_inv @ (b + u)\n",
+ "y_steady = A_inv @ (b_steady + u)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fdcea11c",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "p = B @ y"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "af0eb9ce",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plt.plot(np.arange(0, T)+1, y, label='y')\n",
+ "plt.plot(np.arange(0, T)+1, p, label='p')\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y/p')\n",
+ "plt.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "731dd291",
+ "metadata": {},
+ "source": [
+ "Can you explain why the trend of the price is downward over time?\n",
+ "\n",
+ "Also consider the case when $ y_{0} $ and $ y_{-1} $ are at the\n",
+ "steady state."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fbdcb8a2",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "p_steady = B @ y_steady\n",
+ "\n",
+ "plt.plot(np.arange(0, T)+1, y_steady, label='y')\n",
+ "plt.plot(np.arange(0, T)+1, p_steady, label='p')\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y/p')\n",
+ "plt.legend()\n",
+ "\n",
+ "plt.show()"
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476283.4326148,
+ "filename": "time_series_with_matrices.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Univariate Time Series with Matrix Algebra"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/troubleshooting.ipynb b/_notebooks/troubleshooting.ipynb
new file mode 100644
index 000000000..0b65a2093
--- /dev/null
+++ b/_notebooks/troubleshooting.ipynb
@@ -0,0 +1,94 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "04113743",
+ "metadata": {},
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "77ff69bf",
+ "metadata": {},
+ "source": [
+ "# Troubleshooting\n",
+ "\n",
+ "This page is for readers experiencing errors when running the code from the lectures."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b0916e44",
+ "metadata": {},
+ "source": [
+ "## Fixing your local environment\n",
+ "\n",
+ "The basic assumption of the lectures is that code in a lecture should execute whenever\n",
+ "\n",
+ "1. it is executed in a Jupyter notebook and \n",
+ "1. the notebook is running on a machine with the latest version of Anaconda Python. \n",
+ "\n",
+ "\n",
+ "You have installed Anaconda, haven’t you, following the instructions in [this lecture](https://python-programming.quantecon.org/getting_started.html)?\n",
+ "\n",
+ "Assuming that you have, the most common source of problems for our readers is that their Anaconda distribution is not up to date.\n",
+ "\n",
+ "[Here’s a useful article](https://www.anaconda.com/blog/keeping-anaconda-date)\n",
+ "on how to update Anaconda.\n",
+ "\n",
+ "Another option is to simply remove Anaconda and reinstall.\n",
+ "\n",
+ "You also need to keep the external code libraries, such as [QuantEcon.py](https://quantecon.org/quantecon-py) up to date.\n",
+ "\n",
+ "For this task you can either\n",
+ "\n",
+ "- use conda install -y quantecon on the command line, or \n",
+ "- execute !conda install -y quantecon within a Jupyter notebook. \n",
+ "\n",
+ "\n",
+ "If your local environment is still not working you can do two things.\n",
+ "\n",
+ "First, you can use a remote machine instead, by clicking on the Launch Notebook icon available for each lecture\n",
+ "\n",
+ "\n",
+ "\n",
+ "Second, you can report an issue, so we can try to fix your local set up.\n",
+ "\n",
+ "We like getting feedback on the lectures so please don’t hesitate to get in\n",
+ "touch."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f3cb177d",
+ "metadata": {},
+ "source": [
+ "## Reporting an issue\n",
+ "\n",
+ "One way to give feedback is to raise an issue through our [issue tracker](https://github.com/QuantEcon/lecture-python/issues).\n",
+ "\n",
+ "Please be as specific as possible. Tell us where the problem is and as much\n",
+ "detail about your local set up as you can provide.\n",
+ "\n",
+ "Another feedback option is to use our [discourse forum](https://discourse.quantecon.org/).\n",
+ "\n",
+ "Finally, you can provide direct feedback to [contact@quantecon.org](mailto:contact@quantecon.org)"
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476283.4389536,
+ "filename": "troubleshooting.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Troubleshooting"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/unpleasant.ipynb b/_notebooks/unpleasant.ipynb
new file mode 100644
index 000000000..ba6265311
--- /dev/null
+++ b/_notebooks/unpleasant.ipynb
@@ -0,0 +1,682 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "99326dfd",
+ "metadata": {},
+ "source": [
+ "# Some Unpleasant Monetarist Arithmetic"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5e2cf570",
+ "metadata": {},
+ "source": [
+ "## Overview\n",
+ "\n",
+ "This lecture builds on concepts and issues introduced in [Money Financed Government Deficits and Price Levels](https://intro.quantecon.org/money_inflation.html).\n",
+ "\n",
+ "That lecture describes stationary equilibria that reveal a [*Laffer curve*](https://en.wikipedia.org/wiki/Laffer_curve) in the inflation tax rate and the associated stationary rate of return\n",
+ "on currency.\n",
+ "\n",
+ "In this lecture we study a situation in which a stationary equilibrium prevails after date $ T > 0 $, but not before then.\n",
+ "\n",
+ "For $ t=0, \\ldots, T-1 $, the money supply, price level, and interest-bearing government debt vary along a transition path that ends at $ t=T $.\n",
+ "\n",
+ "During this transition, the ratio of the real balances $ \\frac{m_{t+1}}{{p_t}} $ to indexed one-period government bonds $ \\tilde R B_{t-1} $ maturing at time $ t $ decreases each period.\n",
+ "\n",
+ "This has consequences for the **gross-of-interest** government deficit that must be financed by printing money for times $ t \\geq T $.\n",
+ "\n",
+ "The critical **money-to-bonds** ratio stabilizes only at time $ T $ and afterwards.\n",
+ "\n",
+ "And the larger is $ T $, the higher is the gross-of-interest government deficit that must be financed\n",
+ "by printing money at times $ t \\geq T $.\n",
+ "\n",
+ "These outcomes are the essential finding of Sargent and Wallace’s “unpleasant monetarist arithmetic” [[Sargent and Wallace, 1981](https://intro.quantecon.org/zreferences.html#id293)].\n",
+ "\n",
+ "That lecture described supplies and demands for money that appear in lecture.\n",
+ "\n",
+ "It also characterized the steady state equilibrium from which we work backwards in this lecture.\n",
+ "\n",
+ "In addition to learning about “unpleasant monetarist arithmetic”, in this lecture we’ll learn how to implement a [*fixed point*](https://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29) algorithm for computing an initial price level."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4c174cb7",
+ "metadata": {},
+ "source": [
+ "## Setup\n",
+ "\n",
+ "Let’s start with quick reminders of the model’s components set out in [Money Financed Government Deficits and Price Levels](https://intro.quantecon.org/money_inflation.html).\n",
+ "\n",
+ "Please consult that lecture for more details and Python code that we’ll also use in this lecture.\n",
+ "\n",
+ "For $ t \\geq 1 $, **real balances** evolve according to\n",
+ "\n",
+ "$$\n",
+ "\\frac{m_{t+1}}{p_t} - \\frac{m_{t}}{p_{t-1}} \\frac{p_{t-1}}{p_t} = g\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "b_t - b_{t-1} R_{t-1} = g \\tag{30.1}\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "- $ b_t = \\frac{m_{t+1}}{p_t} $ is real balances at the end of period $ t $ \n",
+ "- $ R_{t-1} = \\frac{p_{t-1}}{p_t} $ is the gross rate of return on real balances held from $ t-1 $ to $ t $ \n",
+ "\n",
+ "\n",
+ "The demand for real balances is\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "b_t = \\gamma_1 - \\gamma_2 R_t^{-1} . \\tag{30.2}\n",
+ "$$\n",
+ "\n",
+ "where $ \\gamma_1 > \\gamma_2 > 0 $."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6f430039",
+ "metadata": {},
+ "source": [
+ "## Monetary-Fiscal Policy\n",
+ "\n",
+ "To the basic model of [Money Financed Government Deficits and Price Levels](https://intro.quantecon.org/money_inflation.html), we add inflation-indexed one-period government bonds as an additional way for the government to finance government expenditures.\n",
+ "\n",
+ "Let $ \\widetilde R > 1 $ be a time-invariant gross real rate of return on government one-period inflation-indexed bonds.\n",
+ "\n",
+ "With this additional source of funds, the government’s budget constraint at time $ t \\geq 0 $ is now\n",
+ "\n",
+ "$$\n",
+ "B_t + \\frac{m_{t+1}}{p_t} = \\widetilde R B_{t-1} + \\frac{m_t}{p_t} + g\n",
+ "$$\n",
+ "\n",
+ "Just before the beginning of time $ 0 $, the public owns $ \\check m_0 $ units of currency (measured in dollars)\n",
+ "and $ \\widetilde R \\check B_{-1} $ units of one-period indexed bonds (measured in time $ 0 $ goods); these two quantities are initial conditions set outside the model.\n",
+ "\n",
+ "Notice that $ \\check m_0 $ is a *nominal* quantity, being measured in dollars, while\n",
+ "$ \\widetilde R \\check B_{-1} $ is a *real* quantity, being measured in time $ 0 $ goods."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "75e4d734",
+ "metadata": {},
+ "source": [
+ "### Open market operations\n",
+ "\n",
+ "At time $ 0 $, government can rearrange its portfolio of debts subject to the following constraint (on open-market operations):\n",
+ "\n",
+ "$$\n",
+ "\\widetilde R B_{-1} + \\frac{m_0}{p_0} = \\widetilde R \\check B_{-1} + \\frac{\\check m_0}{p_0}\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "B_{-1} - \\check B_{-1} = \\frac{1}{p_0 \\widetilde R} \\left( \\check m_0 - m_0 \\right) \\tag{30.3}\n",
+ "$$\n",
+ "\n",
+ "This equation says that the government (e.g., the central bank) can *decrease* $ m_0 $ relative to\n",
+ "$ \\check m_0 $ by *increasing* $ B_{-1} $ relative to $ \\check B_{-1} $.\n",
+ "\n",
+ "This is a version of a standard constraint on a central bank’s [**open market operations**](https://www.federalreserve.gov/monetarypolicy/openmarket.htm) in which it expands the stock of money by buying government bonds from the public."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6e3b528f",
+ "metadata": {},
+ "source": [
+ "## An open market operation at $ t=0 $\n",
+ "\n",
+ "Following Sargent and Wallace [[Sargent and Wallace, 1981](https://intro.quantecon.org/zreferences.html#id293)], we analyze consequences of a central bank policy that\n",
+ "uses an open market operation to lower the price level in the face of a persistent fiscal\n",
+ "deficit that takes the form of a positive $ g $.\n",
+ "\n",
+ "Just before time $ 0 $, the government chooses $ (m_0, B_{-1}) $ subject to constraint\n",
+ "[(30.3)](#equation-eq-openmarketconstraint).\n",
+ "\n",
+ "For $ t =0, 1, \\ldots, T-1 $,\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "B_t & = \\widetilde R B_{t-1} + g \\cr\n",
+ "m_{t+1} & = m_0 \n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "while for $ t \\geq T $,\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "B_t & = B_{T-1} \\cr\n",
+ "m_{t+1} & = m_t + p_t \\overline g\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\overline g = \\left[(\\tilde R -1) B_{T-1} + g \\right] \\tag{30.4}\n",
+ "$$\n",
+ "\n",
+ "We want to compute an equilibrium $ \\{p_t,m_t,b_t, R_t\\}_{t=0} $ sequence under this scheme for\n",
+ "running monetary and fiscal policies.\n",
+ "\n",
+ "Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $ \\{g_t\\}_{t=0}^\\infty $ that must be financed by issuing to the public either money or interest bearing bonds.\n",
+ "\n",
+ "By **monetary policy** or **debt-management policy**, we mean the collection of actions that determine how the government divides its portfolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money).\n",
+ "\n",
+ "By an **open market operation**, we mean a government monetary policy action in which the government\n",
+ "(or its delegate, say, a central bank) either buys government bonds from the public for newly issued money, or sells bonds to the public and withdraws the money it receives from public circulation."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "68bc250f",
+ "metadata": {},
+ "source": [
+ "## Algorithm (basic idea)\n",
+ "\n",
+ "We work backwards from $ t=T $ and first compute $ p_T, R_u $ associated with the low-inflation, low-inflation-tax-rate stationary equilibrium in [Inflation Rate Laffer Curves](https://intro.quantecon.org/money_inflation_nonlinear.html).\n",
+ "\n",
+ "To start our description of our algorithm, it is useful to recall that a stationary rate of return\n",
+ "on currency $ \\bar R $ solves the quadratic equation\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "-\\gamma_2 + (\\gamma_1 + \\gamma_2 - \\overline g) \\bar R - \\gamma_1 \\bar R^2 = 0 \\tag{30.5}\n",
+ "$$\n",
+ "\n",
+ "Quadratic equation [(30.5)](#equation-eq-up-steadyquadratic) has two roots, $ R_l < R_u < 1 $.\n",
+ "\n",
+ "For reasons described at the end of [Money Financed Government Deficits and Price Levels](https://intro.quantecon.org/money_inflation.html), we select the larger root $ R_u $.\n",
+ "\n",
+ "Next, we compute\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "R_T & = R_u \\cr\n",
+ "b_T & = \\gamma_1 - \\gamma_2 R_u^{-1} \\cr\n",
+ "p_T & = \\frac{m_0}{\\gamma_1 - \\overline g - \\gamma_2 R_u^{-1}}\n",
+ "\\end{aligned} \\tag{30.6}\n",
+ "$$\n",
+ "\n",
+ "We can compute continuation sequences $ \\{R_t, b_t\\}_{t=T+1}^\\infty $ of rates of return and real balances that are associated with an equilibrium by solving equation [(30.1)](#equation-eq-up-bmotion) and [(30.2)](#equation-eq-up-bdemand) sequentially for $ t \\geq 1 $:\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "b_t & = b_{t-1} R_{t-1} + \\overline g \\cr\n",
+ "R_t^{-1} & = \\frac{\\gamma_1}{\\gamma_2} - \\gamma_2^{-1} b_t \\cr\n",
+ "p_t & = R_t p_{t-1} \\cr\n",
+ " m_t & = b_{t-1} p_t \n",
+ "\\end{aligned}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "289f22b1",
+ "metadata": {},
+ "source": [
+ "## Before time $ T $\n",
+ "\n",
+ "Define\n",
+ "\n",
+ "$$\n",
+ "\\lambda \\equiv \\frac{\\gamma_2}{\\gamma_1}.\n",
+ "$$\n",
+ "\n",
+ "Our restrictions that $ \\gamma_1 > \\gamma_2 > 0 $ imply that $ \\lambda \\in [0,1) $.\n",
+ "\n",
+ "We want to compute\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "p_0 & = \\gamma_1^{-1} \\left[ \\sum_{j=0}^\\infty \\lambda^j m_{j} \\right] \\cr\n",
+ "& = \\gamma_1^{-1} \\left[ \\sum_{j=0}^{T-1} \\lambda^j m_{0} + \\sum_{j=T}^\\infty \\lambda^j m_{1+j} \\right]\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "Thus,\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "p_0 & = \\gamma_1^{-1} m_0 \\left\\{ \\frac{1 - \\lambda^T}{1-\\lambda} + \\frac{\\lambda^T}{R_u-\\lambda} \\right\\} \\cr\n",
+ "p_1 & = \\gamma_1^{-1} m_0 \\left\\{ \\frac{1 - \\lambda^{T-1}}{1-\\lambda} + \\frac{\\lambda^{T-1}}{R_u-\\lambda} \\right\\} \\cr\n",
+ "\\quad \\vdots & \\quad \\quad \\vdots \\cr\n",
+ "p_{T-1} & = \\gamma_1^{-1} m_0 \\left\\{ \\frac{1 - \\lambda}{1-\\lambda} + \\frac{\\lambda}{R_u-\\lambda} \\right\\} \\cr\n",
+ "p_T & = \\gamma_1^{-1} m_0 \\left\\{\\frac{1}{R_u-\\lambda} \\right\\}\n",
+ "\\end{aligned} \\tag{30.7}\n",
+ "$$\n",
+ "\n",
+ "We can implement the preceding formulas by iterating on\n",
+ "\n",
+ "$$\n",
+ "p_t = \\gamma_1^{-1} m_0 + \\lambda p_{t+1}, \\quad t = T-1, T-2, \\ldots, 0\n",
+ "$$\n",
+ "\n",
+ "starting from\n",
+ "\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "p_T = \\frac{m_0}{\\gamma_1 - \\overline g - \\gamma_2 R_u^{-1}} = \\gamma_1^{-1} m_0 \\left\\{\\frac{1}{R_u-\\lambda} \\right\\} \\tag{30.8}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "fa7f6eb7",
+ "metadata": {},
+ "source": [
+ "## \n",
+ "\n",
+ "We can verify the equivalence of the two formulas on the right sides of [(30.8)](#equation-eq-ptformula) by recalling that\n",
+ "$ R_u $ is a root of the quadratic equation [(30.5)](#equation-eq-up-steadyquadratic) that determines steady state rates of return on currency."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2c6613c5",
+ "metadata": {},
+ "source": [
+ "## Algorithm (pseudo code)\n",
+ "\n",
+ "Now let’s describe a computational algorithm in more detail in the form of a description\n",
+ "that constitutes pseudo code because it approaches a set of instructions we could provide to a\n",
+ "Python coder.\n",
+ "\n",
+ "To compute an equilibrium, we deploy the following algorithm."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0babe520",
+ "metadata": {},
+ "source": [
+ "## \n",
+ "\n",
+ "Given *parameters* include $ g, \\check m_0, \\check B_{-1}, \\widetilde R >1, T $.\n",
+ "\n",
+ "We define a mapping from $ p_0 $ to $ \\widehat p_0 $ as follows.\n",
+ "\n",
+ "- Set $ m_0 $ and then compute $ B_{-1} $ to satisfy the constraint on time $ 0 $ **open market operations** \n",
+ "\n",
+ "\n",
+ "$$\n",
+ "B_{-1}- \\check B_{-1} = \\frac{\\widetilde R}{p_0} \\left( \\check m_0 - m_0 \\right)\n",
+ "$$\n",
+ "\n",
+ "- Compute $ B_{T-1} $ from \n",
+ "\n",
+ "\n",
+ "$$\n",
+ "B_{T-1} = \\widetilde R^T B_{-1} + \\left( \\frac{1 - \\widetilde R^T}{1-\\widetilde R} \\right) g\n",
+ "$$\n",
+ "\n",
+ "- Compute \n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\overline g = g + \\left[ \\tilde R - 1\\right] B_{T-1}\n",
+ "$$\n",
+ "\n",
+ "- Compute $ R_u, p_T $ from formulas [(30.5)](#equation-eq-up-steadyquadratic) and [(30.6)](#equation-eq-laffertstationary) above \n",
+ "- Compute a new estimate of $ p_0 $, call it $ \\widehat p_0 $, from equation [(30.7)](#equation-eq-allts) above \n",
+ "- Note that the preceding steps define a mapping \n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\widehat p_0 = {\\mathcal S}(p_0)\n",
+ "$$\n",
+ "\n",
+ "- We seek a fixed point of $ {\\mathcal S} $, i.e., a solution of $ p_0 = {\\mathcal S}(p_0) $. \n",
+ "- Compute a fixed point by iterating to convergence on the relaxation algorithm \n",
+ "\n",
+ "\n",
+ "$$\n",
+ "p_{0,j+1} = (1-\\theta) {\\mathcal S}(p_{0,j}) + \\theta p_{0,j},\n",
+ "$$\n",
+ "\n",
+ "where $ \\theta \\in [0,1) $ is a relaxation parameter."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4cba7b65",
+ "metadata": {},
+ "source": [
+ "## Example Calculations\n",
+ "\n",
+ "We’ll set parameters of the model so that the steady state after time $ T $ is initially the same\n",
+ "as in [Inflation Rate Laffer Curves](https://intro.quantecon.org/money_inflation_nonlinear.html)\n",
+ "\n",
+ "In particular, we set $ \\gamma_1=100, \\gamma_2 =50, g=3.0 $. We set $ m_0 = 100 $ in that lecture,\n",
+ "but now the counterpart will be $ M_T $, which is endogenous.\n",
+ "\n",
+ "As for new parameters, we’ll set $ \\tilde R = 1.01, \\check B_{-1} = 0, \\check m_0 = 105, T = 5 $.\n",
+ "\n",
+ "We’ll study a “small” open market operation by setting $ m_0 = 100 $.\n",
+ "\n",
+ "These parameter settings mean that just before time $ 0 $, the “central bank” sells the public bonds in exchange for $ \\check m_0 - m_0 = 5 $ units of currency.\n",
+ "\n",
+ "That leaves the public with less currency but more government interest-bearing bonds.\n",
+ "\n",
+ "Since the public has less currency (its supply has diminished) it is plausible to anticipate that the price level at time $ 0 $ will be driven downward.\n",
+ "\n",
+ "But that is not the end of the story, because this **open market operation** at time $ 0 $ has consequences for future settings of $ m_{t+1} $ and the gross-of-interest government deficit $ \\bar g_t $.\n",
+ "\n",
+ "Let’s start with some imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e11703d9",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from collections import namedtuple"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "996d8939",
+ "metadata": {},
+ "source": [
+ "Now let’s dive in and implement our pseudo code in Python."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "33c4caf2",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "# Create a namedtuple that contains parameters\n",
+ "MoneySupplyModel = namedtuple(\"MoneySupplyModel\", \n",
+ " [\"γ1\", \"γ2\", \"g\",\n",
+ " \"R_tilde\", \"m0_check\", \"Bm1_check\",\n",
+ " \"T\"])\n",
+ "\n",
+ "def create_model(γ1=100, γ2=50, g=3.0,\n",
+ " R_tilde=1.01,\n",
+ " Bm1_check=0, m0_check=105,\n",
+ " T=5):\n",
+ " \n",
+ " return MoneySupplyModel(γ1=γ1, γ2=γ2, g=g,\n",
+ " R_tilde=R_tilde,\n",
+ " m0_check=m0_check, Bm1_check=Bm1_check,\n",
+ " T=T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "fd13434b",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "msm = create_model()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "cb2b4afb",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def S(p0, m0, model):\n",
+ "\n",
+ " # unpack parameters\n",
+ " γ1, γ2, g = model.γ1, model.γ2, model.g\n",
+ " R_tilde = model.R_tilde\n",
+ " m0_check, Bm1_check = model.m0_check, model.Bm1_check\n",
+ " T = model.T\n",
+ "\n",
+ " # open market operation\n",
+ " Bm1 = 1 / (p0 * R_tilde) * (m0_check - m0) + Bm1_check\n",
+ "\n",
+ " # compute B_{T-1}\n",
+ " BTm1 = R_tilde ** T * Bm1 + ((1 - R_tilde ** T) / (1 - R_tilde)) * g\n",
+ "\n",
+ " # compute g bar\n",
+ " g_bar = g + (R_tilde - 1) * BTm1\n",
+ "\n",
+ " # solve the quadratic equation\n",
+ " Ru = np.roots((-γ1, γ1 + γ2 - g_bar, -γ2)).max()\n",
+ "\n",
+ " # compute p0\n",
+ " λ = γ2 / γ1\n",
+ " p0_new = (1 / γ1) * m0 * ((1 - λ ** T) / (1 - λ) + λ ** T / (Ru - λ))\n",
+ "\n",
+ " return p0_new"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "05357459",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def compute_fixed_point(m0, p0_guess, model, θ=0.5, tol=1e-6):\n",
+ "\n",
+ " p0 = p0_guess\n",
+ " error = tol + 1\n",
+ "\n",
+ " while error > tol:\n",
+ " p0_next = (1 - θ) * S(p0, m0, model) + θ * p0\n",
+ "\n",
+ " error = np.abs(p0_next - p0)\n",
+ " p0 = p0_next\n",
+ "\n",
+ " return p0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d310db6a",
+ "metadata": {},
+ "source": [
+ "Let’s look at how price level $ p_0 $ in the stationary $ R_u $ equilibrium depends on the initial\n",
+ "money supply $ m_0 $.\n",
+ "\n",
+ "Notice that the slope of $ p_0 $ as a function of $ m_0 $ is constant.\n",
+ "\n",
+ "This outcome indicates that our model verifies a quantity theory of money outcome,\n",
+ "something that Sargent and Wallace [[Sargent and Wallace, 1981](https://intro.quantecon.org/zreferences.html#id293)] purposefully built into their model to justify\n",
+ "the adjective *monetarist* in their title."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e30ca3dc",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "m0_arr = np.arange(10, 110, 10)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f094050f",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plt.plot(m0_arr, [compute_fixed_point(m0, 1, msm) for m0 in m0_arr])\n",
+ "\n",
+ "plt.ylabel('$p_0$')\n",
+ "plt.xlabel('$m_0$')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "179efd10",
+ "metadata": {},
+ "source": [
+ "Now let’s write and implement code that lets us experiment with the time $ 0 $ open market operation described earlier."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7edcbaf9",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def simulate(m0, model, length=15, p0_guess=1):\n",
+ "\n",
+ " # unpack parameters\n",
+ " γ1, γ2, g = model.γ1, model.γ2, model.g\n",
+ " R_tilde = model.R_tilde\n",
+ " m0_check, Bm1_check = model.m0_check, model.Bm1_check\n",
+ " T = model.T\n",
+ "\n",
+ " # (pt, mt, bt, Rt)\n",
+ " paths = np.empty((4, length))\n",
+ "\n",
+ " # open market operation\n",
+ " p0 = compute_fixed_point(m0, 1, model)\n",
+ " Bm1 = 1 / (p0 * R_tilde) * (m0_check - m0) + Bm1_check\n",
+ " BTm1 = R_tilde ** T * Bm1 + ((1 - R_tilde ** T) / (1 - R_tilde)) * g\n",
+ " g_bar = g + (R_tilde - 1) * BTm1\n",
+ " Ru = np.roots((-γ1, γ1 + γ2 - g_bar, -γ2)).max()\n",
+ "\n",
+ " λ = γ2 / γ1\n",
+ "\n",
+ " # t = 0\n",
+ " paths[0, 0] = p0\n",
+ " paths[1, 0] = m0\n",
+ "\n",
+ " # 1 <= t <= T\n",
+ " for t in range(1, T+1, 1):\n",
+ " paths[0, t] = (1 / γ1) * m0 * \\\n",
+ " ((1 - λ ** (T - t)) / (1 - λ)\n",
+ " + (λ ** (T - t) / (Ru - λ)))\n",
+ " paths[1, t] = m0\n",
+ "\n",
+ " # t > T\n",
+ " for t in range(T+1, length):\n",
+ " paths[0, t] = paths[0, t-1] / Ru\n",
+ " paths[1, t] = paths[1, t-1] + paths[0, t] * g_bar\n",
+ "\n",
+ " # Rt = pt / pt+1\n",
+ " paths[3, :T] = paths[0, :T] / paths[0, 1:T+1]\n",
+ " paths[3, T:] = Ru\n",
+ "\n",
+ " # bt = γ1 - γ2 / Rt\n",
+ " paths[2, :] = γ1 - γ2 / paths[3, :]\n",
+ "\n",
+ " return paths"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "610ee267",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "def plot_path(m0_arr, model, length=15):\n",
+ "\n",
+ " fig, axs = plt.subplots(2, 2, figsize=(8, 5))\n",
+ " titles = ['$p_t$', '$m_t$', '$b_t$', '$R_t$']\n",
+ " \n",
+ " for m0 in m0_arr:\n",
+ " paths = simulate(m0, model, length=length)\n",
+ " for i, ax in enumerate(axs.flat):\n",
+ " ax.plot(paths[i])\n",
+ " ax.set_title(titles[i])\n",
+ " \n",
+ " axs[0, 1].hlines(model.m0_check, 0, length, color='r', linestyle='--')\n",
+ " axs[0, 1].text(length * 0.8, model.m0_check * 0.9, r'$\\check{m}_0$')\n",
+ " plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c760d85a",
+ "metadata": {
+ "hide-output": false
+ },
+ "outputs": [],
+ "source": [
+ "plot_path([80, 100], msm)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "34821702",
+ "metadata": {},
+ "source": [
+ "Fig. 30.1 summarizes outcomes of two experiments that convey messages of Sargent and Wallace [[Sargent and Wallace, 1981](https://intro.quantecon.org/zreferences.html#id293)].\n",
+ "\n",
+ "- An open market operation that reduces the supply of money at time $ t=0 $ reduces the price level at time $ t=0 $ \n",
+ "- The lower is the post-open-market-operation money supply at time $ 0 $, lower is the price level at time $ 0 $. \n",
+ "- An open market operation that reduces the post open market operation money supply at time $ 0 $ also *lowers* the rate of return on money $ R_u $ at times $ t \\geq T $ because it brings a higher gross of interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $ t \\geq T $. \n",
+ "- $ R $ is important in the context of maintaining monetary stability and addressing the consequences of increased inflation due to government deficits. Thus, a larger $ R $ might be chosen to mitigate the negative impacts on the real rate of return caused by inflation. "
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476283.4678297,
+ "filename": "unpleasant.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "Some Unpleasant Monetarist Arithmetic"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/_notebooks/zreferences.ipynb b/_notebooks/zreferences.ipynb
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--- /dev/null
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@@ -0,0 +1,234 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "66f1b494",
+ "metadata": {},
+ "source": [
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c08f5570",
+ "metadata": {},
+ "source": [
+ "# References\n",
+ "\n",
+ "\n",
+ "\\[AR02\\] Daron Acemoglu and James A. Robinson. The political economy of the Kuznets curve. *Review of Development Economics*, 6(2):183–203, 2002.\n",
+ "\n",
+ "\n",
+ "\\[AKM+18\\] SeHyoun Ahn, Greg Kaplan, Benjamin Moll, Thomas Winberry, and Christian Wolf. When inequality matters for macro and macro matters for inequality. *NBER Macroeconomics Annual*, 32(1):1–75, 2018.\n",
+ "\n",
+ "\n",
+ "\\[Axt01\\] Robert L Axtell. Zipf distribution of us firm sizes. *science*, 293(5536):1818–1820, 2001.\n",
+ "\n",
+ "\n",
+ "\\[Bar79\\] Robert J Barro. On the Determination of the Public Debt. *Journal of Political Economy*, 87(5):940–971, 1979.\n",
+ "\n",
+ "\n",
+ "\\[BB18\\] Jess Benhabib and Alberto Bisin. Skewed wealth distributions: theory and empirics. *Journal of Economic Literature*, 56(4):1261–91, 2018.\n",
+ "\n",
+ "\n",
+ "\\[BBL19\\] Jess Benhabib, Alberto Bisin, and Mi Luo. Wealth Distribution and Social Mobility in the US: A Quantitative Approach. *American Economic Review*, 109(5):1623–1647, May 2019.\n",
+ "\n",
+ "\n",
+ "\\[Ber97\\] J. N. Bertsimas, D. & Tsitsiklis. *Introduction to linear optimization*. Athena Scientific, 1997.\n",
+ "\n",
+ "\n",
+ "\\[BEGS18\\] Anmol Bhandari, David Evans, Mikhail Golosov, and Thomas J Sargent. Inequality, business cycles, and monetary-fiscal policy. Technical Report, National Bureau of Economic Research, 2018.\n",
+ "\n",
+ "\n",
+ "\\[BEJ18\\] Stephen P Borgatti, Martin G Everett, and Jeffrey C Johnson. *Analyzing social networks*. Sage, 2018.\n",
+ "\n",
+ "\n",
+ "\\[BF90\\] Michael Bruno and Stanley Fischer. Seigniorage, operating rules, and the high inflation trap. *The Quarterly Journal of Economics*, 105(2):353–374, 1990.\n",
+ "\n",
+ "\n",
+ "\\[BW84\\] John Bryant and Neil Wallace. A price discrimination analysis of monetary policy. *The Review of Economic Studies*, 51(2):279–288, 1984.\n",
+ "\n",
+ "\n",
+ "\\[Bur23\\] Jennifer Burns. *Milton Friedman: The Last Conservative by Jennifer Burns*. Farrar, Straus, and Giroux, New York, 2023.\n",
+ "\n",
+ "\n",
+ "\\[Cag56\\] Philip Cagan. The monetary dynamics of hyperinflation. In Milton Friedman, editor, *Studies in the Quantity Theory of Money*, pages 25–117. University of Chicago Press, Chicago, 1956.\n",
+ "\n",
+ "\n",
+ "\\[CB96\\] Marcus J Chambers and Roy E Bailey. A theory of commodity price fluctuations. *Journal of Political Economy*, 104(5):924–957, 1996.\n",
+ "\n",
+ "\n",
+ "\\[Coc23\\] John H Cochrane. *The Fiscal Theory of the Price Level*. Princeton University Press, Princeton, New Jersey, 2023.\n",
+ "\n",
+ "\n",
+ "\\[Cos21\\] Michele Coscia. The atlas for the aspiring network scientist. *arXiv preprint arXiv:2101.00863*, 2021.\n",
+ "\n",
+ "\n",
+ "\\[DL92\\] Angus Deaton and Guy Laroque. On the behavior of commodity prices. *The Review of Economic Studies*, 59:1–23, 1992.\n",
+ "\n",
+ "\n",
+ "\\[DL96\\] Angus Deaton and Guy Laroque. Competitive storage and commodity price dynamics. *Journal of Political Economy*, 104(5):896–923, 1996.\n",
+ "\n",
+ "\n",
+ "\\[DSS58\\] Robert Dorfman, Paul A. Samuelson, and Robert M. Solow. *Linear Programming and Economic Analysis: Revised Edition*. McGraw Hill, New York, 1958.\n",
+ "\n",
+ "\n",
+ "\\[EK+10\\] David Easley, Jon Kleinberg, and others. *Networks, crowds, and markets*. Volume 8. Cambridge university press Cambridge, 2010.\n",
+ "\n",
+ "\n",
+ "\\[Fri56\\] M. Friedman. *A Theory of the Consumption Function*. Princeton University Press, 1956.\n",
+ "\n",
+ "\n",
+ "\\[FK45\\] Milton Friedman and Simon Kuznets. *Income from Independent Professional Practice*. National Bureau of Economic Research, New York, 1945.\n",
+ "\n",
+ "\n",
+ "\\[FDGA+04\\] Yoshi Fujiwara, Corrado Di Guilmi, Hideaki Aoyama, Mauro Gallegati, and Wataru Souma. Do pareto–zipf and gibrat laws hold true? an analysis with european firms. *Physica A: Statistical Mechanics and its Applications*, 335(1-2):197–216, 2004.\n",
+ "\n",
+ "\n",
+ "\\[Gab16\\] Xavier Gabaix. Power laws in economics: an introduction. *Journal of Economic Perspectives*, 30(1):185–206, 2016.\n",
+ "\n",
+ "\n",
+ "\\[GSS03\\] Edward Glaeser, Jose Scheinkman, and Andrei Shleifer. The injustice of inequality. *Journal of Monetary Economics*, 50(1):199–222, 2003.\n",
+ "\n",
+ "\n",
+ "\\[Goy23\\] Sanjeev Goyal. *Networks: An economics approach*. MIT Press, 2023.\n",
+ "\n",
+ "\n",
+ "\\[Hal78\\] Robert E Hall. Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence. *Journal of Political Economy*, 86(6):971–987, 1978.\n",
+ "\n",
+ "\n",
+ "\\[Ham05\\] James D Hamilton. What's real about the business cycle? *Federal Reserve Bank of St. Louis Review*, pages 435–452, 2005.\n",
+ "\n",
+ "\n",
+ "\\[Har60\\] Arthur A. Harlow. The hog cycle and the cobweb theorem. *American Journal of Agricultural Economics*, 42(4):842–853, 1960. [doi:https://doi.org/10.2307/1235116](https://doi.org/https://doi.org/10.2307/1235116).\n",
+ "\n",
+ "\n",
+ "\\[Hu18\\] Y. Hu, Y. & Guo. *Operations research*. Tsinghua University Press, 5th edition, 2018.\n",
+ "\n",
+ "\n",
+ "\\[Haggstrom02\\] Olle Häggström. *Finite Markov chains and algorithmic applications*. Volume 52. Cambridge University Press, 2002.\n",
+ "\n",
+ "\n",
+ "\\[IT23\\] Patrick Imam and Jonathan RW Temple. Political institutions and output collapses. *IMF Working Paper*, 2023.\n",
+ "\n",
+ "\n",
+ "\\[Jac10\\] Matthew O Jackson. *Social and economic networks*. Princeton university press, 2010.\n",
+ "\n",
+ "\n",
+ "\\[Key40\\] John Maynard Keynes. How to pay for the war. In *Essays in persuasion*, pages 367–439. Springer, 1940.\n",
+ "\n",
+ "\n",
+ "\\[KLS18\\] Illenin Kondo, Logan T Lewis, and Andrea Stella. On the us firm and establishment size distributions. Technical Report, SSRN, 2018.\n",
+ "\n",
+ "\n",
+ "\\[KF39\\] Simon Kuznets and Milton Friedman. Incomes from independent professional practice, 1929-1936. *National Bureau of Economic Research Bulletin*, 1939.\n",
+ "\n",
+ "\n",
+ "\\[Lev19\\] Malcolm Levitt. Why did ancient states collapse?: the dysfunctional state. *Why Did Ancient States Collapse?*, pages 1–56, 2019.\n",
+ "\n",
+ "\n",
+ "\\[Man63\\] Benoit Mandelbrot. The variation of certain speculative prices. *The Journal of Business*, 36(4):394–419, 1963.\n",
+ "\n",
+ "\n",
+ "\\[MN03\\] Albert Marcet and Juan P Nicolini. Recurrent hyperinflations and learning. *American Economic Review*, 93(5):1476–1498, 2003.\n",
+ "\n",
+ "\n",
+ "\\[MS89\\] Albert Marcet and Thomas J Sargent. Least squares learning and the dynamics of hyperinflation. In William Barnett, John Geweke and Karl Shell, editors, *Sunspots, Complexity, and Chaos*. Cambridge University Press, 1989.\n",
+ "\n",
+ "\n",
+ "\\[MFD20\\] Filippo Menczer, Santo Fortunato, and Clayton A Davis. *A first course in network science*. Cambridge University Press, 2020.\n",
+ "\n",
+ "\n",
+ "\\[MT09\\] S P Meyn and R L Tweedie. *Markov Chains and Stochastic Stability*. Cambridge University Press, 2009.\n",
+ "\n",
+ "\n",
+ "\\[New18\\] Mark Newman. *Networks*. Oxford university press, 2018.\n",
+ "\n",
+ "\n",
+ "\\[NW89\\] Douglass C North and Barry R Weingast. Constitutions and commitment: the evolution of institutions governing public choice in seventeenth-century england. *The journal of economic history*, 49(4):803–832, 1989.\n",
+ "\n",
+ "\n",
+ "\\[Rac03\\] Svetlozar Todorov Rachev. *Handbook of heavy tailed distributions in finance: Handbooks in finance*. Volume 1. Elsevier, 2003.\n",
+ "\n",
+ "\n",
+ "\\[RRGM11\\] Hernán D Rozenfeld, Diego Rybski, Xavier Gabaix, and Hernán A Makse. The area and population of cities: new insights from a different perspective on cities. *American Economic Review*, 101(5):2205–25, 2011.\n",
+ "\n",
+ "\n",
+ "\\[Rus04\\] Bertrand Russell. *History of western philosophy*. Routledge, 2004.\n",
+ "\n",
+ "\n",
+ "\\[Sam58\\] Paul A Samuelson. An exact consumption-loan model of interest with or without the social contrivance of money. *Journal of political economy*, 66(6):467–482, 1958.\n",
+ "\n",
+ "\n",
+ "\\[Sam71\\] Paul A Samuelson. Stochastic speculative price. *Proceedings of the National Academy of Sciences*, 68(2):335–337, 1971.\n",
+ "\n",
+ "\n",
+ "\\[Sam39\\] Paul A. Samuelson. Interactions between the multiplier analysis and the principle of acceleration. *Review of Economic Studies*, 21(2):75–78, 1939.\n",
+ "\n",
+ "\n",
+ "\\[SWZ09\\] Thomas Sargent, Noah Williams, and Tao Zha. The conquest of south american inflation. *Journal of Political Economy*, 117(2):211–256, 2009.\n",
+ "\n",
+ "\n",
+ "\\[Sar82\\] Thomas J Sargent. The ends of four big inflations. In Robert E Hall, editor, *Inflation: Causes and effects*, pages 41–98. University of Chicago Press, 1982.\n",
+ "\n",
+ "\n",
+ "\\[Sar13\\] Thomas J Sargent. *Rational Expectations and Inflation*. Princeton University Press, Princeton, New Jersey, 2013.\n",
+ "\n",
+ "\n",
+ "\\[SS22\\] Thomas J Sargent and John Stachurski. Economic networks: theory and computation. *arXiv preprint arXiv:2203.11972*, 2022.\n",
+ "\n",
+ "\n",
+ "\\[SS23\\] Thomas J Sargent and John Stachurski. Economic networks: theory and computation. *arXiv preprint arXiv:2203.11972*, 2023.\n",
+ "\n",
+ "\n",
+ "\\[SV95\\] Thomas J Sargent and Francois R Velde. Macroeconomic features of the french revolution. *Journal of Political Economy*, 103(3):474–518, 1995.\n",
+ "\n",
+ "\n",
+ "\\[SV02\\] Thomas J Sargent and François R Velde. *The Big Problem of Small Change*. Princeton University Press, Princeton, New Jersey, 2002.\n",
+ "\n",
+ "\n",
+ "\\[SW81\\] Thomas J Sargent and Neil Wallace. Some unpleasant monetarist arithmetic. *Federal reserve bank of minneapolis quarterly review*, 5(3):1–17, 1981.\n",
+ "\n",
+ "\n",
+ "\\[SS83\\] Jose A Scheinkman and Jack Schechtman. A simple competitive model with production and storage. *The Review of Economic Studies*, 50(3):427–441, 1983.\n",
+ "\n",
+ "\n",
+ "\\[Sch69\\] Thomas C Schelling. Models of Segregation. *American Economic Review*, 59(2):488–493, 1969.\n",
+ "\n",
+ "\n",
+ "\\[ST19\\] Christian Schluter and Mark Trede. Size distributions reconsidered. *Econometric Reviews*, 38(6):695–710, 2019.\n",
+ "\n",
+ "\n",
+ "\\[Smi10\\] Adam Smith. *The Wealth of Nations: An inquiry into the nature and causes of the Wealth of Nations*. Harriman House Limited, 2010.\n",
+ "\n",
+ "\n",
+ "\\[Too14\\] Adam Tooze. The deluge: the great war, america and the remaking of the global order, 1916–1931. 2014.\n",
+ "\n",
+ "\n",
+ "\\[Vil96\\] Pareto Vilfredo. Cours d'économie politique. *Rouge, Lausanne*, 1896.\n",
+ "\n",
+ "\n",
+ "\\[Wau64\\] Frederick V. Waugh. Cobweb models. *Journal of Farm Economics*, 46(4):732–750, 1964.\n",
+ "\n",
+ "\n",
+ "\\[WW82\\] Brian D Wright and Jeffrey C Williams. The economic role of commodity storage. *The Economic Journal*, 92(367):596–614, 1982.\n",
+ "\n",
+ "\n",
+ "\\[Zha12\\] Dongmei Zhao. *Power Distribution and Performance Analysis for Wireless Communication Networks*. SpringerBriefs in Computer Science. Springer US, Boston, MA, 2012. ISBN 978-1-4614-3283-8 978-1-4614-3284-5. URL: [https://link.springer.com/10.1007/978-1-4614-3284-5](https://link.springer.com/10.1007/978-1-4614-3284-5) (visited on 2023-02-03), [doi:10.1007/978-1-4614-3284-5](https://doi.org/10.1007/978-1-4614-3284-5)."
+ ]
+ }
+ ],
+ "metadata": {
+ "date": 1745476283.4873137,
+ "filename": "zreferences.md",
+ "kernelspec": {
+ "display_name": "Python",
+ "language": "python3",
+ "name": "python3"
+ },
+ "title": "References"
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
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+{"index_toc.md": "intro.md"}
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diff --git a/lectures/about.md b/_sources/about.md
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rename to _sources/about.md
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "1573e804",
+ "metadata": {},
+ "source": [
+ "(ar1)=\n",
+ "```{raw} html\n",
+ "\n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "(ar1_processes)=\n",
+ "# AR(1) Processes\n",
+ "\n",
+ "```{index} single: Autoregressive processes\n",
+ "```\n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "In this lecture we are going to study a very simple class of stochastic\n",
+ "models called AR(1) processes.\n",
+ "\n",
+ "These simple models are used again and again in economic research to represent the dynamics of series such as\n",
+ "\n",
+ "* labor income\n",
+ "* dividends\n",
+ "* productivity, etc.\n",
+ "\n",
+ "We are going to study AR(1) processes partly because they are useful and\n",
+ "partly because they help us understand important concepts. \n",
+ "\n",
+ "Let's start with some imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c6c35110",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "plt.rcParams[\"figure.figsize\"] = (11, 5) #set default figure size"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "83fee26c",
+ "metadata": {},
+ "source": [
+ "## The AR(1) model\n",
+ "\n",
+ "The **AR(1) model** (autoregressive model of order 1) takes the form\n",
+ "\n",
+ "```{math}\n",
+ ":label: can_ar1\n",
+ "\n",
+ "X_{t+1} = a X_t + b + c W_{t+1}\n",
+ "```\n",
+ "\n",
+ "where $a, b, c$ are scalar-valued parameters \n",
+ "\n",
+ "(Equation {eq}`can_ar1` is sometimes called a **stochastic difference equation**.)\n",
+ "\n",
+ "```{prf:example}\n",
+ ":label: ar1_ex_ar\n",
+ "\n",
+ "For example, $X_t$ might be \n",
+ "\n",
+ "* the log of labor income for a given household, or\n",
+ "* the log of money demand in a given economy.\n",
+ "\n",
+ "In either case, {eq}`can_ar1` shows that the current value evolves as a linear function\n",
+ "of the previous value and an IID shock $W_{t+1}$.\n",
+ "\n",
+ "(We use $t+1$ for the subscript of $W_{t+1}$ because this random variable is not\n",
+ "observed at time $t$.)\n",
+ "```\n",
+ "\n",
+ "The specification {eq}`can_ar1` generates a time series $\\{ X_t\\}$ as soon as we\n",
+ "specify an initial condition $X_0$.\n",
+ "\n",
+ "To make things even simpler, we will assume that\n",
+ "\n",
+ "* the process $\\{ W_t \\}$ is {ref}`IID \n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "```{index} single: python\n",
+ "```\n",
+ "\n",
+ "# Complex Numbers and Trigonometry\n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "This lecture introduces some elementary mathematics and trigonometry.\n",
+ "\n",
+ "Useful and interesting in its own right, these concepts reap substantial rewards when studying dynamics generated\n",
+ "by linear difference equations or linear differential equations.\n",
+ "\n",
+ "For example, these tools are keys to understanding outcomes attained by Paul\n",
+ "Samuelson (1939) {cite}`Samuelson1939` in his classic paper on interactions\n",
+ "between the investment accelerator and the Keynesian consumption function, our\n",
+ "topic in the lecture {doc}`Samuelson Multiplier Accelerator \n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "```{index} single: python\n",
+ "```\n",
+ "\n",
+ "# Geometric Series for Elementary Economics\n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "The lecture describes important ideas in economics that use the mathematics of geometric series.\n",
+ "\n",
+ "Among these are\n",
+ "\n",
+ "- the Keynesian **multiplier**\n",
+ "- the money **multiplier** that prevails in fractional reserve banking\n",
+ " systems\n",
+ "- interest rates and present values of streams of payouts from assets\n",
+ "\n",
+ "(As we shall see below, the term **multiplier** comes down to meaning **sum of a convergent geometric series**)\n",
+ "\n",
+ "These and other applications prove the truth of the wise crack that\n",
+ "\n",
+ "```{epigraph}\n",
+ "\"In economics, a little knowledge of geometric series goes a long way.\"\n",
+ "```\n",
+ "\n",
+ "Below we'll use the following imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6dc3d2fa",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "plt.rcParams[\"figure.figsize\"] = (11, 5) #set default figure size\n",
+ "import numpy as np\n",
+ "import sympy as sym\n",
+ "from sympy import init_printing\n",
+ "from matplotlib import cm"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a6fbe2e3",
+ "metadata": {},
+ "source": [
+ "## Key formulas\n",
+ "\n",
+ "To start, let $c$ be a real number that lies strictly between\n",
+ "$-1$ and $1$.\n",
+ "\n",
+ "- We often write this as $c \\in (-1,1)$.\n",
+ "- Here $(-1,1)$ denotes the collection of all real numbers that\n",
+ " are strictly less than $1$ and strictly greater than $-1$.\n",
+ "- The symbol $\\in$ means *in* or *belongs to the set after the symbol*.\n",
+ "\n",
+ "We want to evaluate geometric series of two types -- infinite and finite.\n",
+ "\n",
+ "### Infinite geometric series\n",
+ "\n",
+ "The first type of geometric that interests us is the infinite series\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots\n",
+ "$$\n",
+ "\n",
+ "Where $\\cdots$ means that the series continues without end.\n",
+ "\n",
+ "The key formula is\n",
+ "\n",
+ "```{math}\n",
+ ":label: infinite\n",
+ "\n",
+ "1 + c + c^2 + c^3 + \\cdots = \\frac{1}{1 -c }\n",
+ "```\n",
+ "\n",
+ "To prove key formula {eq}`infinite`, multiply both sides by $(1-c)$ and verify\n",
+ "that if $c \\in (-1,1)$, then the outcome is the\n",
+ "equation $1 = 1$.\n",
+ "\n",
+ "### Finite geometric series\n",
+ "\n",
+ "The second series that interests us is the finite geometric series\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots + c^T\n",
+ "$$\n",
+ "\n",
+ "where $T$ is a positive integer.\n",
+ "\n",
+ "The key formula here is\n",
+ "\n",
+ "$$\n",
+ "1 + c + c^2 + c^3 + \\cdots + c^T = \\frac{1 - c^{T+1}}{1-c}\n",
+ "$$\n",
+ "\n",
+ "```{prf:remark}\n",
+ ":label: geom_formula\n",
+ "The above formula works for any value of the scalar\n",
+ "$c$. We don't have to restrict $c$ to be in the\n",
+ "set $(-1,1)$.\n",
+ "```\n",
+ "\n",
+ "We now move on to describe some famous economic applications of\n",
+ "geometric series.\n",
+ "\n",
+ "## Example: The Money Multiplier in Fractional Reserve Banking\n",
+ "\n",
+ "In a fractional reserve banking system, banks hold only a fraction\n",
+ "$r \\in (0,1)$ of cash behind each **deposit receipt** that they\n",
+ "issue\n",
+ "\n",
+ "* In recent times\n",
+ " - cash consists of pieces of paper issued by the government and\n",
+ " called dollars or pounds or $\\ldots$\n",
+ " - a *deposit* is a balance in a checking or savings account that\n",
+ " entitles the owner to ask the bank for immediate payment in cash\n",
+ "* When the UK and France and the US were on either a gold or silver\n",
+ " standard (before 1914, for example)\n",
+ " - cash was a gold or silver coin\n",
+ " - a *deposit receipt* was a *bank note* that the bank promised to\n",
+ " convert into gold or silver on demand; (sometimes it was also a\n",
+ " checking or savings account balance)\n",
+ "\n",
+ "Economists and financiers often define the **supply of money** as an\n",
+ "economy-wide sum of **cash** plus **deposits**.\n",
+ "\n",
+ "In a **fractional reserve banking system** (one in which the reserve\n",
+ "ratio $r$ satisfies $0 < r < 1$), **banks create money** by issuing deposits *backed* by fractional reserves plus loans that they make to their customers.\n",
+ "\n",
+ "A geometric series is a key tool for understanding how banks create\n",
+ "money (i.e., deposits) in a fractional reserve system.\n",
+ "\n",
+ "The geometric series formula {eq}`infinite` is at the heart of the classic model of the money creation process -- one that leads us to the celebrated\n",
+ "**money multiplier**.\n",
+ "\n",
+ "### A simple model\n",
+ "\n",
+ "There is a set of banks named $i = 0, 1, 2, \\ldots$.\n",
+ "\n",
+ "Bank $i$'s loans $L_i$, deposits $D_i$, and\n",
+ "reserves $R_i$ must satisfy the balance sheet equation (because\n",
+ "**balance sheets balance**):\n",
+ "\n",
+ "```{math}\n",
+ ":label: balance\n",
+ "\n",
+ "L_i + R_i = D_i\n",
+ "```\n",
+ "\n",
+ "The left side of the above equation is the sum of the bank's **assets**,\n",
+ "namely, the loans $L_i$ it has outstanding plus its reserves of\n",
+ "cash $R_i$.\n",
+ "\n",
+ "The right side records bank $i$'s liabilities,\n",
+ "namely, the deposits $D_i$ held by its depositors; these are\n",
+ "IOU's from the bank to its depositors in the form of either checking\n",
+ "accounts or savings accounts (or before 1914, bank notes issued by a\n",
+ "bank stating promises to redeem notes for gold or silver on demand).\n",
+ "\n",
+ "Each bank $i$ sets its reserves to satisfy the equation\n",
+ "\n",
+ "```{math}\n",
+ ":label: reserves\n",
+ "\n",
+ "R_i = r D_i\n",
+ "```\n",
+ "\n",
+ "where $r \\in (0,1)$ is its **reserve-deposit ratio** or **reserve\n",
+ "ratio** for short\n",
+ "\n",
+ "- the reserve ratio is either set by a government or chosen by banks\n",
+ " for precautionary reasons\n",
+ "\n",
+ "Next we add a theory stating that bank $i+1$'s deposits depend\n",
+ "entirely on loans made by bank $i$, namely\n",
+ "\n",
+ "```{math}\n",
+ ":label: deposits\n",
+ "\n",
+ "D_{i+1} = L_i\n",
+ "```\n",
+ "\n",
+ "Thus, we can think of the banks as being arranged along a line with\n",
+ "loans from bank $i$ being immediately deposited in $i+1$\n",
+ "\n",
+ "- in this way, the debtors to bank $i$ become creditors of\n",
+ " bank $i+1$\n",
+ "\n",
+ "Finally, we add an *initial condition* about an exogenous level of bank\n",
+ "$0$'s deposits\n",
+ "\n",
+ "$$\n",
+ "D_0 \\ \\text{ is given exogenously}\n",
+ "$$\n",
+ "\n",
+ "We can think of $D_0$ as being the amount of cash that a first\n",
+ "depositor put into the first bank in the system, bank number $i=0$.\n",
+ "\n",
+ "Now we do a little algebra.\n",
+ "\n",
+ "Combining equations {eq}`balance` and {eq}`reserves` tells us that\n",
+ "\n",
+ "```{math}\n",
+ ":label: fraction\n",
+ "\n",
+ "L_i = (1-r) D_i\n",
+ "```\n",
+ "\n",
+ "This states that bank $i$ loans a fraction $(1-r)$ of its\n",
+ "deposits and keeps a fraction $r$ as cash reserves.\n",
+ "\n",
+ "Combining equation {eq}`fraction` with equation {eq}`deposits` tells us that\n",
+ "\n",
+ "$$\n",
+ "D_{i+1} = (1-r) D_i \\ \\text{ for } i \\geq 0\n",
+ "$$\n",
+ "\n",
+ "which implies that\n",
+ "\n",
+ "```{math}\n",
+ ":label: geomseries\n",
+ "\n",
+ "D_i = (1 - r)^i D_0 \\ \\text{ for } i \\geq 0\n",
+ "```\n",
+ "\n",
+ "Equation {eq}`geomseries` expresses $D_i$ as the $i$ th term in the\n",
+ "product of $D_0$ and the geometric series\n",
+ "\n",
+ "$$\n",
+ "1, (1-r), (1-r)^2, \\cdots\n",
+ "$$\n",
+ "\n",
+ "Therefore, the sum of all deposits in our banking system\n",
+ "$i=0, 1, 2, \\ldots$ is\n",
+ "\n",
+ "```{math}\n",
+ ":label: sumdeposits\n",
+ "\n",
+ "\\sum_{i=0}^\\infty (1-r)^i D_0 = \\frac{D_0}{1 - (1-r)} = \\frac{D_0}{r}\n",
+ "```\n",
+ "\n",
+ "### Money multiplier\n",
+ "\n",
+ "The **money multiplier** is a number that tells the multiplicative\n",
+ "factor by which an exogenous injection of cash into bank $0$ leads\n",
+ "to an increase in the total deposits in the banking system.\n",
+ "\n",
+ "Equation {eq}`sumdeposits` asserts that the **money multiplier** is\n",
+ "$\\frac{1}{r}$\n",
+ "\n",
+ "- An initial deposit of cash of $D_0$ in bank $0$ leads\n",
+ " the banking system to create total deposits of $\\frac{D_0}{r}$.\n",
+ "- The initial deposit $D_0$ is held as reserves, distributed\n",
+ " throughout the banking system according to $D_0 = \\sum_{i=0}^\\infty R_i$.\n",
+ "\n",
+ "## Example: The Keynesian Multiplier\n",
+ "\n",
+ "The famous economist John Maynard Keynes and his followers created a\n",
+ "simple model intended to determine national income $y$ in\n",
+ "circumstances in which\n",
+ "\n",
+ "- there are substantial unemployed resources, in particular **excess\n",
+ " supply** of labor and capital\n",
+ "- prices and interest rates fail to adjust to make aggregate **supply\n",
+ " equal demand** (e.g., prices and interest rates are frozen)\n",
+ "- national income is entirely determined by aggregate demand\n",
+ "\n",
+ "### Static version\n",
+ "\n",
+ "An elementary Keynesian model of national income determination consists\n",
+ "of three equations that describe aggregate demand for $y$ and its\n",
+ "components.\n",
+ "\n",
+ "The first equation is a national income identity asserting that\n",
+ "consumption $c$ plus investment $i$ equals national income\n",
+ "$y$:\n",
+ "\n",
+ "$$\n",
+ "c+ i = y\n",
+ "$$\n",
+ "\n",
+ "The second equation is a Keynesian consumption function asserting that\n",
+ "people consume a fraction $b \\in (0,1)$ of their income:\n",
+ "\n",
+ "$$\n",
+ "c = b y\n",
+ "$$\n",
+ "\n",
+ "The fraction $b \\in (0,1)$ is called the **marginal propensity to\n",
+ "consume**.\n",
+ "\n",
+ "The fraction $1-b \\in (0,1)$ is called the **marginal propensity\n",
+ "to save**.\n",
+ "\n",
+ "The third equation simply states that investment is exogenous at level\n",
+ "$i$.\n",
+ "\n",
+ "- *exogenous* means *determined outside this model*.\n",
+ "\n",
+ "Substituting the second equation into the first gives $(1-b) y = i$.\n",
+ "\n",
+ "Solving this equation for $y$ gives\n",
+ "\n",
+ "$$\n",
+ "y = \\frac{1}{1-b} i\n",
+ "$$\n",
+ "\n",
+ "The quantity $\\frac{1}{1-b}$ is called the **investment\n",
+ "multiplier** or simply the **multiplier**.\n",
+ "\n",
+ "Applying the formula for the sum of an infinite geometric series, we can\n",
+ "write the above equation as\n",
+ "\n",
+ "$$\n",
+ "y = i \\sum_{t=0}^\\infty b^t\n",
+ "$$\n",
+ "\n",
+ "where $t$ is a nonnegative integer.\n",
+ "\n",
+ "So we arrive at the following equivalent expressions for the multiplier:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{1-b} = \\sum_{t=0}^\\infty b^t\n",
+ "$$\n",
+ "\n",
+ "The expression $\\sum_{t=0}^\\infty b^t$ motivates an interpretation\n",
+ "of the multiplier as the outcome of a dynamic process that we describe\n",
+ "next.\n",
+ "\n",
+ "### Dynamic version\n",
+ "\n",
+ "We arrive at a dynamic version by interpreting the nonnegative integer\n",
+ "$t$ as indexing time and changing our specification of the\n",
+ "consumption function to take time into account\n",
+ "\n",
+ "- we add a one-period lag in how income affects consumption\n",
+ "\n",
+ "We let $c_t$ be consumption at time $t$ and $i_t$ be\n",
+ "investment at time $t$.\n",
+ "\n",
+ "We modify our consumption function to assume the form\n",
+ "\n",
+ "$$\n",
+ "c_t = b y_{t-1}\n",
+ "$$\n",
+ "\n",
+ "so that $b$ is the marginal propensity to consume (now) out of\n",
+ "last period's income.\n",
+ "\n",
+ "We begin with an initial condition stating that\n",
+ "\n",
+ "$$\n",
+ "y_{-1} = 0\n",
+ "$$\n",
+ "\n",
+ "We also assume that\n",
+ "\n",
+ "$$\n",
+ "i_t = i \\ \\ \\textrm {for all } t \\geq 0\n",
+ "$$\n",
+ "\n",
+ "so that investment is constant over time.\n",
+ "\n",
+ "It follows that\n",
+ "\n",
+ "$$\n",
+ "y_0 = i + c_0 = i + b y_{-1} = i\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "$$\n",
+ "y_1 = c_1 + i = b y_0 + i = (1 + b) i\n",
+ "$$\n",
+ "\n",
+ "and\n",
+ "\n",
+ "$$\n",
+ "y_2 = c_2 + i = b y_1 + i = (1 + b + b^2) i\n",
+ "$$\n",
+ "\n",
+ "and more generally\n",
+ "\n",
+ "$$\n",
+ "y_t = b y_{t-1} + i = (1+ b + b^2 + \\cdots + b^t) i\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "y_t = \\frac{1-b^{t+1}}{1 -b } i\n",
+ "$$\n",
+ "\n",
+ "Evidently, as $t \\rightarrow + \\infty$,\n",
+ "\n",
+ "$$\n",
+ "y_t \\rightarrow \\frac{1}{1-b} i\n",
+ "$$\n",
+ "\n",
+ "**Remark 1:** The above formula is often applied to assert that an\n",
+ "exogenous increase in investment of $\\Delta i$ at time $0$\n",
+ "ignites a dynamic process of increases in national income by successive amounts\n",
+ "\n",
+ "$$\n",
+ "\\Delta i, (1 + b )\\Delta i, (1+b + b^2) \\Delta i , \\cdots\n",
+ "$$\n",
+ "\n",
+ "at times $0, 1, 2, \\ldots$.\n",
+ "\n",
+ "**Remark 2** Let $g_t$ be an exogenous sequence of government\n",
+ "expenditures.\n",
+ "\n",
+ "If we generalize the model so that the national income identity\n",
+ "becomes\n",
+ "\n",
+ "$$\n",
+ "c_t + i_t + g_t = y_t\n",
+ "$$\n",
+ "\n",
+ "then a version of the preceding argument shows that the **government\n",
+ "expenditures multiplier** is also $\\frac{1}{1-b}$, so that a\n",
+ "permanent increase in government expenditures ultimately leads to an\n",
+ "increase in national income equal to the multiplier times the increase\n",
+ "in government expenditures.\n",
+ "\n",
+ "## Example: Interest Rates and Present Values\n",
+ "\n",
+ "We can apply our formula for geometric series to study how interest\n",
+ "rates affect values of streams of dollar payments that extend over time.\n",
+ "\n",
+ "We work in discrete time and assume that $t = 0, 1, 2, \\ldots$\n",
+ "indexes time.\n",
+ "\n",
+ "We let $r \\in (0,1)$ be a one-period **net nominal interest rate**\n",
+ "\n",
+ "- if the nominal interest rate is $5$ percent,\n",
+ " then $r= .05$\n",
+ "\n",
+ "A one-period **gross nominal interest rate** $R$ is defined as\n",
+ "\n",
+ "$$\n",
+ "R = 1 + r \\in (1, 2)\n",
+ "$$\n",
+ "\n",
+ "- if $r=.05$, then $R = 1.05$\n",
+ "\n",
+ "**Remark:** The gross nominal interest rate $R$ is an **exchange\n",
+ "rate** or **relative price** of dollars at between times $t$ and\n",
+ "$t+1$. The units of $R$ are dollars at time $t+1$ per\n",
+ "dollar at time $t$.\n",
+ "\n",
+ "When people borrow and lend, they trade dollars now for dollars later or\n",
+ "dollars later for dollars now.\n",
+ "\n",
+ "The price at which these exchanges occur is the gross nominal interest\n",
+ "rate.\n",
+ "\n",
+ "- If I sell $x$ dollars to you today, you pay me $R x$\n",
+ " dollars tomorrow.\n",
+ "- This means that you borrowed $x$ dollars for me at a gross\n",
+ " interest rate $R$ and a net interest rate $r$.\n",
+ "\n",
+ "We assume that the net nominal interest rate $r$ is fixed over\n",
+ "time, so that $R$ is the gross nominal interest rate at times\n",
+ "$t=0, 1, 2, \\ldots$.\n",
+ "\n",
+ "Two important geometric sequences are\n",
+ "\n",
+ "```{math}\n",
+ ":label: geom1\n",
+ "\n",
+ "1, R, R^2, \\cdots\n",
+ "```\n",
+ "\n",
+ "and\n",
+ "\n",
+ "```{math}\n",
+ ":label: geom2\n",
+ "\n",
+ "1, R^{-1}, R^{-2}, \\cdots\n",
+ "```\n",
+ "\n",
+ "Sequence {eq}`geom1` tells us how dollar values of an investment **accumulate**\n",
+ "through time.\n",
+ "\n",
+ "Sequence {eq}`geom2` tells us how to **discount** future dollars to get their\n",
+ "values in terms of today's dollars.\n",
+ "\n",
+ "### Accumulation\n",
+ "\n",
+ "Geometric sequence {eq}`geom1` tells us how one dollar invested and re-invested\n",
+ "in a project with gross one period nominal rate of return accumulates\n",
+ "\n",
+ "- here we assume that net interest payments are reinvested in the\n",
+ " project\n",
+ "- thus, $1$ dollar invested at time $0$ pays interest\n",
+ " $r$ dollars after one period, so we have $r+1 = R$\n",
+ " dollars at time$1$\n",
+ "- at time $1$ we reinvest $1+r =R$ dollars and receive interest\n",
+ " of $r R$ dollars at time $2$ plus the *principal*\n",
+ " $R$ dollars, so we receive $r R + R = (1+r)R = R^2$\n",
+ " dollars at the end of period $2$\n",
+ "- and so on\n",
+ "\n",
+ "Evidently, if we invest $x$ dollars at time $0$ and\n",
+ "reinvest the proceeds, then the sequence\n",
+ "\n",
+ "$$\n",
+ "x , xR , x R^2, \\cdots\n",
+ "$$\n",
+ "\n",
+ "tells how our account accumulates at dates $t=0, 1, 2, \\ldots$.\n",
+ "\n",
+ "### Discounting\n",
+ "\n",
+ "Geometric sequence {eq}`geom2` tells us how much future dollars are worth in terms of today's dollars.\n",
+ "\n",
+ "Remember that the units of $R$ are dollars at $t+1$ per\n",
+ "dollar at $t$.\n",
+ "\n",
+ "It follows that\n",
+ "\n",
+ "- the units of $R^{-1}$ are dollars at $t$ per dollar at $t+1$\n",
+ "- the units of $R^{-2}$ are dollars at $t$ per dollar at $t+2$\n",
+ "- and so on; the units of $R^{-j}$ are dollars at $t$ per\n",
+ " dollar at $t+j$\n",
+ "\n",
+ "So if someone has a claim on $x$ dollars at time $t+j$, it\n",
+ "is worth $x R^{-j}$ dollars at time $t$ (e.g., today).\n",
+ "\n",
+ "### Application to asset pricing\n",
+ "\n",
+ "A **lease** requires a payments stream of $x_t$ dollars at\n",
+ "times $t = 0, 1, 2, \\ldots$ where\n",
+ "\n",
+ "$$\n",
+ "x_t = G^t x_0\n",
+ "$$\n",
+ "\n",
+ "where $G = (1+g)$ and $g \\in (0,1)$.\n",
+ "\n",
+ "Thus, lease payments increase at $g$ percent per period.\n",
+ "\n",
+ "For a reason soon to be revealed, we assume that $G < R$.\n",
+ "\n",
+ "The **present value** of the lease is\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned} p_0 & = x_0 + x_1/R + x_2/(R^2) + \\cdots \\\\\n",
+ " & = x_0 (1 + G R^{-1} + G^2 R^{-2} + \\cdots ) \\\\\n",
+ " & = x_0 \\frac{1}{1 - G R^{-1}} \\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "where the last line uses the formula for an infinite geometric series.\n",
+ "\n",
+ "Recall that $R = 1+r$ and $G = 1+g$ and that $R > G$\n",
+ "and $r > g$ and that $r$ and $g$ are typically small\n",
+ "numbers, e.g., .05 or .03.\n",
+ "\n",
+ "Use the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) of $\\frac{1}{1+r}$ about $r=0$,\n",
+ "namely,\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{1+r} = 1 - r + r^2 - r^3 + \\cdots\n",
+ "$$\n",
+ "\n",
+ "and the fact that $r$ is small to approximate\n",
+ "$\\frac{1}{1+r} \\approx 1 - r$.\n",
+ "\n",
+ "Use this approximation to write $p_0$ as\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ " p_0 &= x_0 \\frac{1}{1 - G R^{-1}} \\\\\n",
+ " &= x_0 \\frac{1}{1 - (1+g) (1-r) } \\\\\n",
+ " &= x_0 \\frac{1}{1 - (1+g - r - rg)} \\\\\n",
+ " & \\approx x_0 \\frac{1}{r -g }\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "where the last step uses the approximation $r g \\approx 0$.\n",
+ "\n",
+ "The approximation\n",
+ "\n",
+ "$$\n",
+ "p_0 = \\frac{x_0 }{r -g }\n",
+ "$$\n",
+ "\n",
+ "is known as the **Gordon formula** for the present value or current\n",
+ "price of an infinite payment stream $x_0 G^t$ when the nominal\n",
+ "one-period interest rate is $r$ and when $r > g$.\n",
+ "\n",
+ "We can also extend the asset pricing formula so that it applies to finite leases.\n",
+ "\n",
+ "Let the payment stream on the lease now be $x_t$ for $t= 1,2, \\dots,T$, where again\n",
+ "\n",
+ "$$\n",
+ "x_t = G^t x_0\n",
+ "$$\n",
+ "\n",
+ "The present value of this lease is:\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned} \\begin{split}p_0&=x_0 + x_1/R + \\dots +x_T/R^T \\\\ &= x_0(1+GR^{-1}+\\dots +G^{T}R^{-T}) \\\\ &= \\frac{x_0(1-G^{T+1}R^{-(T+1)})}{1-GR^{-1}} \\end{split}\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "Applying the Taylor series to $R^{-(T+1)}$ about $r=0$ we get:\n",
+ "\n",
+ "$$\n",
+ "\\frac{1}{(1+r)^{T+1}}= 1-r(T+1)+\\frac{1}{2}r^2(T+1)(T+2)+\\dots \\approx 1-r(T+1)\n",
+ "$$\n",
+ "\n",
+ "Similarly, applying the Taylor series to $G^{T+1}$ about $g=0$:\n",
+ "\n",
+ "$$\n",
+ "(1+g)^{T+1} = 1+(T+1)g+\\frac{T(T+1)}{2!}g^2+\\frac{(T-1)T(T+1)}{3!}g^3+\\dots \\approx 1+ (T+1)g\n",
+ "$$\n",
+ "\n",
+ "Thus, we get the following approximation:\n",
+ "\n",
+ "$$\n",
+ "p_0 =\\frac{x_0(1-(1+(T+1)g)(1-r(T+1)))}{1-(1-r)(1+g) }\n",
+ "$$\n",
+ "\n",
+ "Expanding:\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned} p_0 &=\\frac{x_0(1-1+(T+1)^2 rg +r(T+1)-g(T+1))}{1-1+r-g+rg} \\\\&=\\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\\\ &= \\frac{x_0(T+1)(r-g)}{r-g + rg}+\\frac{x_0rg(T+1)^2}{r-g+rg}\\\\ &\\approx \\frac{x_0(T+1)(r-g)}{r-g}+\\frac{x_0rg(T+1)}{r-g}\\\\ &= x_0(T+1) + \\frac{x_0rg(T+1)}{r-g} \\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "We could have also approximated by removing the second term\n",
+ "$rgx_0(T+1)$ when $T$ is relatively small compared to\n",
+ "$1/(rg)$ to get $x_0(T+1)$ as in the finite stream\n",
+ "approximation.\n",
+ "\n",
+ "We will plot the true finite stream present-value and the two\n",
+ "approximations, under different values of $T$, and $g$ and $r$ in Python.\n",
+ "\n",
+ "First we plot the true finite stream present-value after computing it\n",
+ "below"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "26fc674e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# True present value of a finite lease\n",
+ "def finite_lease_pv_true(T, g, r, x_0):\n",
+ " G = (1 + g)\n",
+ " R = (1 + r)\n",
+ " return (x_0 * (1 - G**(T + 1) * R**(-T - 1))) / (1 - G * R**(-1))\n",
+ "# First approximation for our finite lease\n",
+ "\n",
+ "def finite_lease_pv_approx_1(T, g, r, x_0):\n",
+ " p = x_0 * (T + 1) + x_0 * r * g * (T + 1) / (r - g)\n",
+ " return p\n",
+ "\n",
+ "# Second approximation for our finite lease\n",
+ "def finite_lease_pv_approx_2(T, g, r, x_0):\n",
+ " return (x_0 * (T + 1))\n",
+ "\n",
+ "# Infinite lease\n",
+ "def infinite_lease(g, r, x_0):\n",
+ " G = (1 + g)\n",
+ " R = (1 + r)\n",
+ " return x_0 / (1 - G * R**(-1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "afb4d644",
+ "metadata": {},
+ "source": [
+ "Now that we have defined our functions, we can plot some outcomes.\n",
+ "\n",
+ "First we study the quality of our approximations"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "71725924",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Finite lease present value $T$ periods ahead",
+ "name": "finite_lease_present_value"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "def plot_function(axes, x_vals, func, args):\n",
+ " axes.plot(x_vals, func(*args), label=func.__name__)\n",
+ "\n",
+ "T_max = 50\n",
+ "\n",
+ "T = np.arange(0, T_max+1)\n",
+ "g = 0.02\n",
+ "r = 0.03\n",
+ "x_0 = 1\n",
+ "\n",
+ "our_args = (T, g, r, x_0)\n",
+ "funcs = [finite_lease_pv_true,\n",
+ " finite_lease_pv_approx_1,\n",
+ " finite_lease_pv_approx_2]\n",
+ " # the three functions we want to compare\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "for f in funcs:\n",
+ " plot_function(ax, T, f, our_args)\n",
+ "ax.legend()\n",
+ "ax.set_xlabel('$T$ Periods Ahead')\n",
+ "ax.set_ylabel('Present Value, $p_0$')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "e6a8a77e",
+ "metadata": {},
+ "source": [
+ "Evidently our approximations perform well for small values of $T$.\n",
+ "\n",
+ "However, holding $g$ and r fixed, our approximations deteriorate as $T$ increases.\n",
+ "\n",
+ "Next we compare the infinite and finite duration lease present values\n",
+ "over different lease lengths $T$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b98490eb",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Infinite and finite lease present value $T$ periods ahead",
+ "name": "infinite_and_finite_lease_present_value"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "# Convergence of infinite and finite\n",
+ "T_max = 1000\n",
+ "T = np.arange(0, T_max+1)\n",
+ "fig, ax = plt.subplots()\n",
+ "f_1 = finite_lease_pv_true(T, g, r, x_0)\n",
+ "f_2 = np.full(T_max+1, infinite_lease(g, r, x_0))\n",
+ "ax.plot(T, f_1, label='T-period lease PV')\n",
+ "ax.plot(T, f_2, '--', label='Infinite lease PV')\n",
+ "ax.set_xlabel('$T$ Periods Ahead')\n",
+ "ax.set_ylabel('Present Value, $p_0$')\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c97af98f",
+ "metadata": {},
+ "source": [
+ "The graph above shows how as duration $T \\rightarrow +\\infty$,\n",
+ "the value of a lease of duration $T$ approaches the value of a\n",
+ "perpetual lease.\n",
+ "\n",
+ "Now we consider two different views of what happens as $r$ and\n",
+ "$g$ covary"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a5f445be",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Value of lease of length $T$",
+ "name": "value_of_lease"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "# First view\n",
+ "# Changing r and g\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.set_ylabel('Present Value, $p_0$')\n",
+ "ax.set_xlabel('$T$ periods ahead')\n",
+ "T_max = 10\n",
+ "T=np.arange(0, T_max+1)\n",
+ "\n",
+ "rs, gs = (0.9, 0.5, 0.4001, 0.4), (0.4, 0.4, 0.4, 0.5),\n",
+ "comparisons = (r'$\\gg$', '$>$', r'$\\approx$', '$<$')\n",
+ "for r, g, comp in zip(rs, gs, comparisons):\n",
+ " ax.plot(finite_lease_pv_true(T, g, r, x_0), label=f'r(={r}) {comp} g(={g})')\n",
+ "\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cc7a5b3e",
+ "metadata": {},
+ "source": [
+ "This graph gives a big hint for why the condition $r > g$ is\n",
+ "necessary if a lease of length $T = +\\infty$ is to have finite\n",
+ "value.\n",
+ "\n",
+ "For fans of 3-d graphs the same point comes through in the following\n",
+ "graph.\n",
+ "\n",
+ "If you aren't enamored of 3-d graphs, feel free to skip the next\n",
+ "visualization!"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e4ddfbc8",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Three period lease PV with varying $g$ and $r$",
+ "name": "three_period_lease_PV"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "# Second view\n",
+ "fig = plt.figure(figsize = [16, 5])\n",
+ "T = 3\n",
+ "ax = plt.subplot(projection='3d')\n",
+ "r = np.arange(0.01, 0.99, 0.005)\n",
+ "g = np.arange(0.011, 0.991, 0.005)\n",
+ "\n",
+ "rr, gg = np.meshgrid(r, g)\n",
+ "z = finite_lease_pv_true(T, gg, rr, x_0)\n",
+ "\n",
+ "# Removes points where undefined\n",
+ "same = (rr == gg)\n",
+ "z[same] = np.nan\n",
+ "surf = ax.plot_surface(rr, gg, z, cmap=cm.coolwarm,\n",
+ " antialiased=True, clim=(0, 15))\n",
+ "fig.colorbar(surf, shrink=0.5, aspect=5)\n",
+ "ax.set_xlabel('$r$')\n",
+ "ax.set_ylabel('$g$')\n",
+ "ax.set_zlabel('Present Value, $p_0$')\n",
+ "ax.view_init(20, 8)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "57f33cde",
+ "metadata": {},
+ "source": [
+ "We can use a little calculus to study how the present value $p_0$\n",
+ "of a lease varies with $r$ and $g$.\n",
+ "\n",
+ "We will use a library called [SymPy](https://www.sympy.org/).\n",
+ "\n",
+ "SymPy enables us to do symbolic math calculations including\n",
+ "computing derivatives of algebraic equations.\n",
+ "\n",
+ "We will illustrate how it works by creating a symbolic expression that\n",
+ "represents our present value formula for an infinite lease.\n",
+ "\n",
+ "After that, we'll use SymPy to compute derivatives"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c210b751",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Creates algebraic symbols that can be used in an algebraic expression\n",
+ "g, r, x0 = sym.symbols('g, r, x0')\n",
+ "G = (1 + g)\n",
+ "R = (1 + r)\n",
+ "p0 = x0 / (1 - G * R**(-1))\n",
+ "init_printing(use_latex='mathjax')\n",
+ "print('Our formula is:')\n",
+ "p0"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c4c84f8c",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print('dp0 / dg is:')\n",
+ "dp_dg = sym.diff(p0, g)\n",
+ "dp_dg"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1aac3b11",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "print('dp0 / dr is:')\n",
+ "dp_dr = sym.diff(p0, r)\n",
+ "dp_dr"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "56b45068",
+ "metadata": {},
+ "source": [
+ "We can see that for $\\frac{\\partial p_0}{\\partial r}<0$ as long as\n",
+ "$r>g$, $r>0$ and $g>0$ and $x_0$ is positive,\n",
+ "so $\\frac{\\partial p_0}{\\partial r}$ will always be negative.\n",
+ "\n",
+ "Similarly, $\\frac{\\partial p_0}{\\partial g}>0$ as long as $r>g$, $r>0$ and $g>0$ and $x_0$ is positive, so $\\frac{\\partial p_0}{\\partial g}$\n",
+ "will always be positive.\n",
+ "\n",
+ "## Back to the Keynesian multiplier\n",
+ "\n",
+ "We will now go back to the case of the Keynesian multiplier and plot the\n",
+ "time path of $y_t$, given that consumption is a constant fraction\n",
+ "of national income, and investment is fixed."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c4c700da",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Path of aggregate output tver time",
+ "name": "path_of_aggregate_output_over_time"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "# Function that calculates a path of y\n",
+ "def calculate_y(i, b, g, T, y_init):\n",
+ " y = np.zeros(T+1)\n",
+ " y[0] = i + b * y_init + g\n",
+ " for t in range(1, T+1):\n",
+ " y[t] = b * y[t-1] + i + g\n",
+ " return y\n",
+ "\n",
+ "# Initial values\n",
+ "i_0 = 0.3\n",
+ "g_0 = 0.3\n",
+ "# 2/3 of income goes towards consumption\n",
+ "b = 2/3\n",
+ "y_init = 0\n",
+ "T = 100\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "ax.set_xlabel('$t$')\n",
+ "ax.set_ylabel('$y_t$')\n",
+ "ax.plot(np.arange(0, T+1), calculate_y(i_0, b, g_0, T, y_init))\n",
+ "# Output predicted by geometric series\n",
+ "ax.hlines(i_0 / (1 - b) + g_0 / (1 - b), xmin=-1, xmax=101, linestyles='--')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3ccb5957",
+ "metadata": {},
+ "source": [
+ "In this model, income grows over time, until it gradually converges to\n",
+ "the infinite geometric series sum of income.\n",
+ "\n",
+ "We now examine what will\n",
+ "happen if we vary the so-called **marginal propensity to consume**,\n",
+ "i.e., the fraction of income that is consumed"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c8002adf",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Changing consumption as a fraction of income",
+ "name": "changing_consumption_as_fraction_of_income"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "bs = (1/3, 2/3, 5/6, 0.9)\n",
+ "\n",
+ "fig,ax = plt.subplots()\n",
+ "ax.set_ylabel('$y_t$')\n",
+ "ax.set_xlabel('$t$')\n",
+ "x = np.arange(0, T+1)\n",
+ "for b in bs:\n",
+ " y = calculate_y(i_0, b, g_0, T, y_init)\n",
+ " ax.plot(x, y, label=r'$b=$'+f\"{b:.2f}\")\n",
+ "ax.legend()\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5e95edf4",
+ "metadata": {},
+ "source": [
+ "Increasing the marginal propensity to consume $b$ increases the\n",
+ "path of output over time.\n",
+ "\n",
+ "Now we will compare the effects on output of increases in investment and government spending."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "698ad1c6",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Different increase on output",
+ "name": "different_increase_on_output"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(6, 10))\n",
+ "fig.subplots_adjust(hspace=0.3)\n",
+ "\n",
+ "x = np.arange(0, T+1)\n",
+ "values = [0.3, 0.4]\n",
+ "\n",
+ "for i in values:\n",
+ " y = calculate_y(i, b, g_0, T, y_init)\n",
+ " ax1.plot(x, y, label=f\"i={i}\")\n",
+ "for g in values:\n",
+ " y = calculate_y(i_0, b, g, T, y_init)\n",
+ " ax2.plot(x, y, label=f\"g={g}\")\n",
+ "\n",
+ "axes = ax1, ax2\n",
+ "param_labels = \"Investment\", \"Government Spending\"\n",
+ "for ax, param in zip(axes, param_labels):\n",
+ " ax.set_title(f'An Increase in {param} on Output')\n",
+ " ax.legend(loc =\"lower right\")\n",
+ " ax.set_ylabel('$y_t$')\n",
+ " ax.set_xlabel('$t$')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "86550007",
+ "metadata": {},
+ "source": [
+ "Notice here, whether government spending increases from 0.3 to 0.4 or\n",
+ "investment increases from 0.3 to 0.4, the shifts in the graphs are\n",
+ "identical."
+ ]
+ }
+ ],
+ "metadata": {
+ "jupytext": {
+ "text_representation": {
+ "extension": ".md",
+ "format_name": "myst",
+ "format_version": 0.13,
+ "jupytext_version": "1.14.5"
+ }
+ },
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
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+ },
+ "source_map": [
+ 12,
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+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
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diff --git a/lectures/geom_series.md b/_sources/geom_series.md
similarity index 100%
rename from lectures/geom_series.md
rename to _sources/geom_series.md
diff --git a/_sources/greek_square.ipynb b/_sources/greek_square.ipynb
new file mode 100644
index 000000000..edc5ec633
--- /dev/null
+++ b/_sources/greek_square.ipynb
@@ -0,0 +1,983 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "ed92e06a",
+ "metadata": {},
+ "source": [
+ "# Computing Square Roots\n",
+ "\n",
+ "\n",
+ "## Introduction\n",
+ "\n",
+ "Chapter 24 of {cite}`russell2004history` about early Greek mathematics and astronomy contains this\n",
+ "fascinating passage:\n",
+ "\n",
+ " ```{epigraph} \n",
+ " The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value were discovered. The best was as follows: Form two columns of numbers, which we will call the $a$'s and the $b$'s; each starts with a $1$. The next $a$, at each stage, is formed by adding the last $a$ and the $b$ already obtained; the next $b$ is formed by adding twice the previous $a$ to the previous $b$. The first 6 pairs so obtained are $(1,1), (2,3), (5,7), (12,17), (29,41), (70,99)$. In each pair, $2 a^2 - b^2$ is $1$ or $-1$. Thus $b/a$ is nearly the square root of two, and at each fresh step it gets nearer. For instance, the reader may satisy himself that the square of $99/70$ is very nearly equal to $2$.\n",
+ " ```\n",
+ "\n",
+ "This lecture drills down and studies this ancient method for computing square roots by using some of the matrix algebra that we've learned in earlier quantecon lectures. \n",
+ "\n",
+ "In particular, this lecture can be viewed as a sequel to {doc}`eigen_I`.\n",
+ "\n",
+ "It provides an example of how eigenvectors isolate *invariant subspaces* that help construct and analyze solutions of linear difference equations. \n",
+ "\n",
+ "When vector $x_t$ starts in an invariant subspace, iterating the different equation keeps $x_{t+j}$\n",
+ "in that subspace for all $j \\geq 1$. \n",
+ "\n",
+ "Invariant subspace methods are used throughout applied economic dynamics, for example, in the lecture {doc}`money_inflation`.\n",
+ "\n",
+ "Our approach here is to illustrate the method with an ancient example, one that ancient Greek mathematicians used to compute square roots of positive integers.\n",
+ "\n",
+ "## Perfect squares and irrational numbers\n",
+ "\n",
+ "An integer is called a **perfect square** if its square root is also an integer.\n",
+ "\n",
+ "An ordered sequence of perfect squares starts with \n",
+ "\n",
+ "$$\n",
+ "4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, \\ldots \n",
+ "$$\n",
+ "\n",
+ "If an integer is not a perfect square, then its square root is an irrational number -- i.e., it cannot be expressed as a ratio of two integers, and its decimal expansion is indefinite.\n",
+ "\n",
+ "The ancient Greeks invented an algorithm to compute square roots of integers, including integers that are not perfect squares.\n",
+ "\n",
+ "Their method involved\n",
+ "\n",
+ " * computing a particular sequence of integers $\\{y_t\\}_{t=0}^\\infty$;\n",
+ " \n",
+ " * computing $\\lim_{t \\rightarrow \\infty} \\left(\\frac{y_{t+1}}{y_t}\\right) = \\bar r$;\n",
+ " \n",
+ " * deducing the desired square root from $\\bar r$.\n",
+ " \n",
+ "In this lecture, we'll describe this method.\n",
+ "\n",
+ "We'll also use invariant subspaces to describe variations on this method that are faster.\n",
+ "\n",
+ "## Second-order linear difference equations\n",
+ "\n",
+ "Before telling how the ancient Greeks computed square roots, we'll provide a quick introduction\n",
+ "to second-order linear difference equations.\n",
+ "\n",
+ "We'll study the following second-order linear difference equation\n",
+ "\n",
+ "$$\n",
+ "y_t = a_1 y_{t-1} + a_2 y_{t-2}, \\quad t \\geq 0\n",
+ "$$ (eq:2diff1)\n",
+ "\n",
+ "where $(y_{-1}, y_{-2})$ is a pair of given initial conditions. \n",
+ "\n",
+ "Equation {eq}`eq:2diff1` is actually an infinite number of linear equations in the sequence\n",
+ "$\\{y_t\\}_{t=0}^\\infty$.\n",
+ "\n",
+ "There is one equation each for $t = 0, 1, 2, \\ldots$. \n",
+ "\n",
+ "We could follow an approach taken in the lecture on {doc}`present values\n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "(scalar_dynam)=\n",
+ "# Dynamics in One Dimension\n",
+ "\n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "In economics many variables depend on their past values\n",
+ "\n",
+ "For example, it seems reasonable to believe that inflation last year with affects inflation this year.\n",
+ "\n",
+ "(Perhaps high inflation last year will lead people to demand higher wages to\n",
+ "compensate, which will feed into higher prices this year.)\n",
+ "\n",
+ "Letting $\\pi_t$ be inflation this year and $\\pi_{t-1}$ be inflation last year, we\n",
+ "can write this relationship in a general form as\n",
+ "\n",
+ "$$ \\pi_t = f(\\pi_{t-1}) $$\n",
+ "\n",
+ "where $f$ is some function describing the relationship between the variables.\n",
+ "\n",
+ "This equation is an example of one-dimensional discrete time dynamic system.\n",
+ "\n",
+ "In this lecture we cover the foundations of one-dimensional discrete time\n",
+ "dynamics.\n",
+ "\n",
+ "(While most quantitative models have two or more state variables, the\n",
+ "one-dimensional setting is a good place to learn foundations \n",
+ "and understand key concepts.)\n",
+ "\n",
+ "Let's start with some standard imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "05f35996",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "import numpy as np"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bc493fa0",
+ "metadata": {},
+ "source": [
+ "## Some definitions\n",
+ "\n",
+ "This section sets out the objects of interest and the kinds of properties we study.\n",
+ "\n",
+ "### Composition of functions\n",
+ "\n",
+ "For this lecture you should know the following.\n",
+ "\n",
+ "If \n",
+ "\n",
+ "* $g$ is a function from $A$ to $B$ and\n",
+ "* $f$ is a function from $B$ to $C$, \n",
+ "\n",
+ "then the **composition** $f \\circ g$ of $f$ and $g$ is defined by\n",
+ "\n",
+ "$$ \n",
+ " (f \\circ g)(x) = f(g(x))\n",
+ "$$\n",
+ "\n",
+ "For example, if \n",
+ "\n",
+ "* $A=B=C=\\mathbb R$, the set of real numbers, \n",
+ "* $g(x)=x^2$ and $f(x)=\\sqrt{x}$, then $(f \\circ g)(x) = \\sqrt{x^2} = |x|$.\n",
+ "\n",
+ "If $f$ is a function from $A$ to itself, then $f^2$ is the composition of $f$\n",
+ "with itself.\n",
+ "\n",
+ "For example, if $A = (0, \\infty)$, the set of positive numbers, and $f(x) =\n",
+ "\\sqrt{x}$, then \n",
+ "\n",
+ "$$\n",
+ " f^2(x) = \\sqrt{\\sqrt{x}} = x^{1/4}\n",
+ "$$\n",
+ "\n",
+ "Similarly, if $n$ is a positive integer, then $f^n$ is $n$ compositions of $f$ with\n",
+ "itself.\n",
+ "\n",
+ "In the example above, $f^n(x) = x^{1/(2^n)}$.\n",
+ "\n",
+ "\n",
+ "\n",
+ "### Dynamic systems\n",
+ "\n",
+ "A **(discrete time) dynamic system** is a set $S$ and a function $g$ that sends\n",
+ "set $S$ back into to itself.\n",
+ "\n",
+ "\n",
+ "Examples of dynamic systems include\n",
+ "\n",
+ "* $S = (0, 1)$ and $g(x) = \\sqrt{x}$\n",
+ "* $S = (0, 1)$ and $g(x) = x^2$\n",
+ "* $S = \\mathbb Z$ (the integers) and $g(x) = 2 x$\n",
+ "\n",
+ "\n",
+ "On the other hand, if $S = (-1, 1)$ and $g(x) = x+1$, then $S$ and $g$ do not\n",
+ "form a dynamic system, since $g(1) = 2$.\n",
+ "\n",
+ "* $g$ does not always send points in $S$ back into $S$.\n",
+ "\n",
+ "We care about dynamic systems because we can use them to study dynamics!\n",
+ "\n",
+ "Given a dynamic system consisting of set $S$ and function $g$, we can create\n",
+ "a sequence $\\{x_t\\}$ of points in $S$ by setting\n",
+ "\n",
+ "```{math}\n",
+ ":label: sdsod\n",
+ " x_{t+1} = g(x_t)\n",
+ " \\quad \\text{ with } \n",
+ " x_0 \\text{ given}.\n",
+ "```\n",
+ "\n",
+ "This means that we choose some number $x_0$ in $S$ and then take\n",
+ "\n",
+ "```{math}\n",
+ ":label: sdstraj\n",
+ " x_0, \\quad\n",
+ " x_1 = g(x_0), \\quad\n",
+ " x_2 = g(x_1) = g(g(x_0)), \\quad \\text{etc.}\n",
+ "```\n",
+ "\n",
+ "This sequence $\\{x_t\\}$ is called the **trajectory** of $x_0$ under $g$.\n",
+ "\n",
+ "In this setting, $S$ is called the **state space** and $x_t$ is called the\n",
+ "**state variable**.\n",
+ "\n",
+ "Recalling that $g^n$ is the $n$ compositions of $g$ with itself, \n",
+ "we can write the trajectory more simply as \n",
+ "\n",
+ "$$\n",
+ " x_t = g^t(x_0) \\quad \\text{ for } t = 0, 1, 2, \\ldots\n",
+ "$$\n",
+ "\n",
+ "In all of what follows, we are going to assume that $S$ is a subset of\n",
+ "$\\mathbb R$, the real numbers.\n",
+ "\n",
+ "Equation {eq}`sdsod` is sometimes called a **first order difference equation**\n",
+ "\n",
+ "* first order means dependence on only one lag (i.e., earlier states such as $x_{t-1}$ do not enter into {eq}`sdsod`).\n",
+ "\n",
+ "\n",
+ "\n",
+ "### Example: a linear model\n",
+ "\n",
+ "One simple example of a dynamic system is when $S=\\mathbb R$ and $g(x)=ax +\n",
+ "b$, where $a, b$ are constants (sometimes called ``parameters'').\n",
+ "\n",
+ "This leads to the **linear difference equation**\n",
+ "\n",
+ "$$\n",
+ " x_{t+1} = a x_t + b \n",
+ " \\quad \\text{ with } \n",
+ " x_0 \\text{ given}.\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "The trajectory of $x_0$ is \n",
+ "\n",
+ "```{math}\n",
+ ":label: sdslinmodpath\n",
+ "\n",
+ "x_0, \\quad\n",
+ "a x_0 + b, \\quad\n",
+ "a^2 x_0 + a b + b, \\quad \\text{etc.}\n",
+ "```\n",
+ "\n",
+ "Continuing in this way, and using our knowledge of {doc}`geometric series\n",
+ "\n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "# Racial Segregation\n",
+ "\n",
+ "```{index} single: Schelling Segregation Model\n",
+ "```\n",
+ "\n",
+ "```{index} single: Models; Schelling's Segregation Model\n",
+ "```\n",
+ "\n",
+ "## Outline\n",
+ "\n",
+ "In 1969, Thomas C. Schelling developed a simple but striking model of racial\n",
+ "segregation {cite}`Schelling1969`.\n",
+ "\n",
+ "His model studies the dynamics of racially mixed neighborhoods.\n",
+ "\n",
+ "Like much of Schelling's work, the model shows how local interactions can lead\n",
+ "to surprising aggregate outcomes.\n",
+ "\n",
+ "It studies a setting where agents (think of households) have relatively mild\n",
+ "preference for neighbors of the same race.\n",
+ "\n",
+ "For example, these agents might be comfortable with a mixed race neighborhood\n",
+ "but uncomfortable when they feel \"surrounded\" by people from a different race.\n",
+ "\n",
+ "Schelling illustrated the follow surprising result: in such a setting, mixed\n",
+ "race neighborhoods are likely to be unstable, tending to collapse over time.\n",
+ "\n",
+ "In fact the model predicts strongly divided neighborhoods, with high levels of\n",
+ "segregation.\n",
+ "\n",
+ "In other words, extreme segregation outcomes arise even though people's\n",
+ "preferences are not particularly extreme.\n",
+ "\n",
+ "These extreme outcomes happen because of *interactions* between agents in the\n",
+ "model (e.g., households in a city) that drive self-reinforcing dynamics in the\n",
+ "model.\n",
+ "\n",
+ "These ideas will become clearer as the lecture unfolds.\n",
+ "\n",
+ "In recognition of his work on segregation and other research, Schelling was\n",
+ "awarded the 2005 Nobel Prize in Economic Sciences (joint with Robert Aumann).\n",
+ "\n",
+ "\n",
+ "Let's start with some imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "721ec459",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import matplotlib.pyplot as plt\n",
+ "from random import uniform, seed\n",
+ "from math import sqrt\n",
+ "import numpy as np"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b7e8ce55",
+ "metadata": {},
+ "source": [
+ "## The model\n",
+ "\n",
+ "In this section we will build a version of Schelling's model.\n",
+ "\n",
+ "### Set-Up\n",
+ "\n",
+ "We will cover a variation of Schelling's model that is different from the\n",
+ "original but also easy to program and, at the same time, captures his main\n",
+ "idea.\n",
+ "\n",
+ "Suppose we have two types of people: orange people and green people.\n",
+ "\n",
+ "Assume there are $n$ of each type.\n",
+ "\n",
+ "These agents all live on a single unit square.\n",
+ "\n",
+ "Thus, the location (e.g, address) of an agent is just a point $(x, y)$, where\n",
+ "$0 < x, y < 1$.\n",
+ "\n",
+ "* The set of all points $(x,y)$ satisfying $0 < x, y < 1$ is called the **unit square**\n",
+ "* Below we denote the unit square by $S$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4264e424",
+ "metadata": {},
+ "source": [
+ "### Preferences\n",
+ "\n",
+ "We will say that an agent is *happy* if 5 or more of her 10 nearest neighbors are of the same type.\n",
+ "\n",
+ "An agent who is not happy is called *unhappy*.\n",
+ "\n",
+ "For example,\n",
+ "\n",
+ "* if an agent is orange and 5 of her 10 nearest neighbors are orange, then she is happy.\n",
+ "* if an agent is green and 8 of her 10 nearest neighbors are orange, then she is unhappy.\n",
+ "\n",
+ "'Nearest' is in terms of [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance).\n",
+ "\n",
+ "An important point to note is that agents are **not** averse to living in mixed areas.\n",
+ "\n",
+ "They are perfectly happy if half of their neighbors are of the other color."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "693504ae",
+ "metadata": {},
+ "source": [
+ "### Behavior\n",
+ "\n",
+ "Initially, agents are mixed together (integrated).\n",
+ "\n",
+ "In particular, we assume that the initial location of each agent is an\n",
+ "independent draw from a bivariate uniform distribution on the unit square $S$.\n",
+ "\n",
+ "* First their $x$ coordinate is drawn from a uniform distribution on $(0,1)$\n",
+ "* Then, independently, their $y$ coordinate is drawn from the same distribution.\n",
+ "\n",
+ "Now, cycling through the set of all agents, each agent is now given the chance to stay or move.\n",
+ "\n",
+ "Each agent stays if they are happy and moves if they are unhappy.\n",
+ "\n",
+ "The algorithm for moving is as follows\n",
+ "\n",
+ "```{prf:algorithm} Jump Chain Algorithm\n",
+ ":label: move_algo\n",
+ "\n",
+ "1. Draw a random location in $S$\n",
+ "1. If happy at new location, move there\n",
+ "1. Otherwise, go to step 1\n",
+ "\n",
+ "```\n",
+ "\n",
+ "We cycle continuously through the agents, each time allowing an unhappy agent\n",
+ "to move.\n",
+ "\n",
+ "We continue to cycle until no one wishes to move."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "eb309bab",
+ "metadata": {},
+ "source": [
+ "## Results\n",
+ "\n",
+ "Let's now implement and run this simulation.\n",
+ "\n",
+ "In what follows, agents are modeled as [objects](https://python-programming.quantecon.org/python_oop.html).\n",
+ "\n",
+ "Here's an indication of their structure:\n",
+ "\n",
+ "```{code-block} none\n",
+ "* Data:\n",
+ "\n",
+ " * type (green or orange)\n",
+ " * location\n",
+ "\n",
+ "* Methods:\n",
+ "\n",
+ " * determine whether happy or not given locations of other agents\n",
+ " * If not happy, move\n",
+ " * find a new location where happy\n",
+ "```\n",
+ "\n",
+ "Let's build them."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1e162cc0",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "class Agent:\n",
+ "\n",
+ " def __init__(self, type):\n",
+ " self.type = type\n",
+ " self.draw_location()\n",
+ "\n",
+ " def draw_location(self):\n",
+ " self.location = uniform(0, 1), uniform(0, 1)\n",
+ "\n",
+ " def get_distance(self, other):\n",
+ " \"Computes the euclidean distance between self and other agent.\"\n",
+ " a = (self.location[0] - other.location[0])**2\n",
+ " b = (self.location[1] - other.location[1])**2\n",
+ " return sqrt(a + b)\n",
+ "\n",
+ " def happy(self,\n",
+ " agents, # List of other agents\n",
+ " num_neighbors=10, # No. of agents viewed as neighbors\n",
+ " require_same_type=5): # How many neighbors must be same type\n",
+ " \"\"\"\n",
+ " True if a sufficient number of nearest neighbors are of the same\n",
+ " type.\n",
+ " \"\"\"\n",
+ "\n",
+ " distances = []\n",
+ "\n",
+ " # Distances is a list of pairs (d, agent), where d is distance from\n",
+ " # agent to self\n",
+ " for agent in agents:\n",
+ " if self != agent:\n",
+ " distance = self.get_distance(agent)\n",
+ " distances.append((distance, agent))\n",
+ "\n",
+ " # Sort from smallest to largest, according to distance\n",
+ " distances.sort()\n",
+ "\n",
+ " # Extract the neighboring agents\n",
+ " neighbors = [agent for d, agent in distances[:num_neighbors]]\n",
+ "\n",
+ " # Count how many neighbors have the same type as self\n",
+ " num_same_type = sum(self.type == agent.type for agent in neighbors)\n",
+ " return num_same_type >= require_same_type\n",
+ "\n",
+ " def update(self, agents):\n",
+ " \"If not happy, then randomly choose new locations until happy.\"\n",
+ " while not self.happy(agents):\n",
+ " self.draw_location()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3f3a2449",
+ "metadata": {},
+ "source": [
+ "Here's some code that takes a list of agents and produces a plot showing their\n",
+ "locations on the unit square.\n",
+ "\n",
+ "Orange agents are represented by orange dots and green ones are represented by\n",
+ "green dots."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "25ab1e6e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def plot_distribution(agents, cycle_num):\n",
+ " \"Plot the distribution of agents after cycle_num rounds of the loop.\"\n",
+ " x_values_0, y_values_0 = [], []\n",
+ " x_values_1, y_values_1 = [], []\n",
+ " # == Obtain locations of each type == #\n",
+ " for agent in agents:\n",
+ " x, y = agent.location\n",
+ " if agent.type == 0:\n",
+ " x_values_0.append(x)\n",
+ " y_values_0.append(y)\n",
+ " else:\n",
+ " x_values_1.append(x)\n",
+ " y_values_1.append(y)\n",
+ " fig, ax = plt.subplots()\n",
+ " plot_args = {'markersize': 8, 'alpha': 0.8}\n",
+ " ax.set_facecolor('azure')\n",
+ " ax.plot(x_values_0, y_values_0,\n",
+ " 'o', markerfacecolor='orange', **plot_args)\n",
+ " ax.plot(x_values_1, y_values_1,\n",
+ " 'o', markerfacecolor='green', **plot_args)\n",
+ " ax.set_title(f'Cycle {cycle_num-1}')\n",
+ " plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0ab4cfcd",
+ "metadata": {},
+ "source": [
+ "And here's some pseudocode for the main loop, where we cycle through the\n",
+ "agents until no one wishes to move.\n",
+ "\n",
+ "The pseudocode is\n",
+ "\n",
+ "```{code-block} none\n",
+ "plot the distribution\n",
+ "while agents are still moving\n",
+ " for agent in agents\n",
+ " give agent the opportunity to move\n",
+ "plot the distribution\n",
+ "```\n",
+ "\n",
+ "The real code is below"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7d7c2674",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def run_simulation(num_of_type_0=600,\n",
+ " num_of_type_1=600,\n",
+ " max_iter=100_000, # Maximum number of iterations\n",
+ " set_seed=1234):\n",
+ "\n",
+ " # Set the seed for reproducibility\n",
+ " seed(set_seed)\n",
+ "\n",
+ " # Create a list of agents of type 0\n",
+ " agents = [Agent(0) for i in range(num_of_type_0)]\n",
+ " # Append a list of agents of type 1\n",
+ " agents.extend(Agent(1) for i in range(num_of_type_1))\n",
+ "\n",
+ " # Initialize a counter\n",
+ " count = 1\n",
+ "\n",
+ " # Plot the initial distribution\n",
+ " plot_distribution(agents, count)\n",
+ "\n",
+ " # Loop until no agent wishes to move\n",
+ " while count < max_iter:\n",
+ " print('Entering loop ', count)\n",
+ " count += 1\n",
+ " no_one_moved = True\n",
+ " for agent in agents:\n",
+ " old_location = agent.location\n",
+ " agent.update(agents)\n",
+ " if agent.location != old_location:\n",
+ " no_one_moved = False\n",
+ " if no_one_moved:\n",
+ " break\n",
+ "\n",
+ " # Plot final distribution\n",
+ " plot_distribution(agents, count)\n",
+ "\n",
+ " if count < max_iter:\n",
+ " print(f'Converged after {count} iterations.')\n",
+ " else:\n",
+ " print('Hit iteration bound and terminated.')\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "946bbf51",
+ "metadata": {},
+ "source": [
+ "Let's have a look at the results."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "866d2d42",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "run_simulation()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9c7e9a58",
+ "metadata": {},
+ "source": [
+ "As discussed above, agents are initially mixed randomly together.\n",
+ "\n",
+ "But after several cycles, they become segregated into distinct regions.\n",
+ "\n",
+ "In this instance, the program terminated after a small number of cycles\n",
+ "through the set of agents, indicating that all agents had reached a state of\n",
+ "happiness.\n",
+ "\n",
+ "What is striking about the pictures is how rapidly racial integration breaks down.\n",
+ "\n",
+ "This is despite the fact that people in the model don't actually mind living mixed with the other type.\n",
+ "\n",
+ "Even with these preferences, the outcome is a high degree of segregation.\n",
+ "\n",
+ "\n",
+ "\n",
+ "## Exercises\n",
+ "\n",
+ "```{exercise-start}\n",
+ ":label: schelling_ex1\n",
+ "```\n",
+ "\n",
+ "The object oriented style that we used for coding above is neat but harder to\n",
+ "optimize than procedural code (i.e., code based around functions rather than\n",
+ "objects and methods).\n",
+ "\n",
+ "Try writing a new version of the model that stores\n",
+ "\n",
+ "* the locations of all agents as a 2D NumPy array of floats.\n",
+ "* the types of all agents as a flat NumPy array of integers.\n",
+ "\n",
+ "Write functions that act on this data to update the model using the logic\n",
+ "similar to that described above.\n",
+ "\n",
+ "However, implement the following two changes:\n",
+ "\n",
+ "1. Agents are offered a move at random (i.e., selected randomly and given the\n",
+ " opportunity to move).\n",
+ "2. After an agent has moved, flip their type with probability 0.01\n",
+ "\n",
+ "The second change introduces extra randomness into the model.\n",
+ "\n",
+ "(We can imagine that, every so often, an agent moves to a different city and,\n",
+ "with small probability, is replaced by an agent of the other type.)\n",
+ "\n",
+ "```{exercise-end}\n",
+ "```\n",
+ "\n",
+ "```{solution-start} schelling_ex1\n",
+ ":class: dropdown\n",
+ "```\n",
+ "solution here"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "71cec659",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from numpy.random import uniform, randint\n",
+ "\n",
+ "n = 1000 # number of agents (agents = 0, ..., n-1)\n",
+ "k = 10 # number of agents regarded as neighbors\n",
+ "require_same_type = 5 # want >= require_same_type neighbors of the same type\n",
+ "\n",
+ "def initialize_state():\n",
+ " locations = uniform(size=(n, 2))\n",
+ " types = randint(0, high=2, size=n) # label zero or one\n",
+ " return locations, types\n",
+ "\n",
+ "\n",
+ "def compute_distances_from_loc(loc, locations):\n",
+ " \"\"\" Compute distance from location loc to all other points. \"\"\"\n",
+ " return np.linalg.norm(loc - locations, axis=1)\n",
+ "\n",
+ "def get_neighbors(loc, locations):\n",
+ " \" Get all neighbors of a given location. \"\n",
+ " all_distances = compute_distances_from_loc(loc, locations)\n",
+ " indices = np.argsort(all_distances) # sort agents by distance to loc\n",
+ " neighbors = indices[:k] # keep the k closest ones\n",
+ " return neighbors\n",
+ "\n",
+ "def is_happy(i, locations, types):\n",
+ " happy = True\n",
+ " agent_loc = locations[i, :]\n",
+ " agent_type = types[i]\n",
+ " neighbors = get_neighbors(agent_loc, locations)\n",
+ " neighbor_types = types[neighbors]\n",
+ " if sum(neighbor_types == agent_type) < require_same_type:\n",
+ " happy = False\n",
+ " return happy\n",
+ "\n",
+ "def count_happy(locations, types):\n",
+ " \" Count the number of happy agents. \"\n",
+ " happy_sum = 0\n",
+ " for i in range(n):\n",
+ " happy_sum += is_happy(i, locations, types)\n",
+ " return happy_sum\n",
+ "\n",
+ "def update_agent(i, locations, types):\n",
+ " \" Move agent if unhappy. \"\n",
+ " moved = False\n",
+ " while not is_happy(i, locations, types):\n",
+ " moved = True\n",
+ " locations[i, :] = uniform(), uniform()\n",
+ " return moved\n",
+ "\n",
+ "def plot_distribution(locations, types, title, savepdf=False):\n",
+ " \" Plot the distribution of agents after cycle_num rounds of the loop.\"\n",
+ " fig, ax = plt.subplots()\n",
+ " colors = 'orange', 'green'\n",
+ " for agent_type, color in zip((0, 1), colors):\n",
+ " idx = (types == agent_type)\n",
+ " ax.plot(locations[idx, 0],\n",
+ " locations[idx, 1],\n",
+ " 'o',\n",
+ " markersize=8,\n",
+ " markerfacecolor=color,\n",
+ " alpha=0.8)\n",
+ " ax.set_title(title)\n",
+ " plt.show()\n",
+ "\n",
+ "def sim_random_select(max_iter=100_000, flip_prob=0.01, test_freq=10_000):\n",
+ " \"\"\"\n",
+ " Simulate by randomly selecting one household at each update.\n",
+ "\n",
+ " Flip the color of the household with probability `flip_prob`.\n",
+ "\n",
+ " \"\"\"\n",
+ "\n",
+ " locations, types = initialize_state()\n",
+ " current_iter = 0\n",
+ "\n",
+ " while current_iter <= max_iter:\n",
+ "\n",
+ " # Choose a random agent and update them\n",
+ " i = randint(0, n)\n",
+ " moved = update_agent(i, locations, types)\n",
+ "\n",
+ " if flip_prob > 0:\n",
+ " # flip agent i's type with probability epsilon\n",
+ " U = uniform()\n",
+ " if U < flip_prob:\n",
+ " current_type = types[i]\n",
+ " types[i] = 0 if current_type == 1 else 1\n",
+ "\n",
+ " # Every so many updates, plot and test for convergence\n",
+ " if current_iter % test_freq == 0:\n",
+ " cycle = current_iter / n\n",
+ " plot_distribution(locations, types, f'iteration {current_iter}')\n",
+ " if count_happy(locations, types) == n:\n",
+ " print(f\"Converged at iteration {current_iter}\")\n",
+ " break\n",
+ "\n",
+ " current_iter += 1\n",
+ "\n",
+ " if current_iter > max_iter:\n",
+ " print(f\"Terminating at iteration {current_iter}\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "93c56d34",
+ "metadata": {},
+ "source": [
+ "```{solution-end}\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "55d96c7a",
+ "metadata": {},
+ "source": [
+ "When we run this we again find that mixed neighborhoods break down and segregation emerges.\n",
+ "\n",
+ "Here's a sample run."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "892739f1",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "sim_random_select(max_iter=50_000, flip_prob=0.01, test_freq=10_000)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "0dc889b1",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "jupytext": {
+ "text_representation": {
+ "extension": ".md",
+ "format_name": "myst",
+ "format_version": 0.13,
+ "jupytext_version": "1.14.1"
+ }
+ },
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "source_map": [
+ 12,
+ 68,
+ 73,
+ 97,
+ 116,
+ 148,
+ 173,
+ 221,
+ 229,
+ 252,
+ 269,
+ 310,
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+ 371,
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+ 482,
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+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
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\ No newline at end of file
diff --git a/lectures/schelling.md b/_sources/schelling.md
similarity index 100%
rename from lectures/schelling.md
rename to _sources/schelling.md
diff --git a/_sources/short_path.ipynb b/_sources/short_path.ipynb
new file mode 100644
index 000000000..ed8f22922
--- /dev/null
+++ b/_sources/short_path.ipynb
@@ -0,0 +1,619 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "8c583c5c",
+ "metadata": {},
+ "source": [
+ "(short_path)=\n",
+ "```{raw} html\n",
+ "\n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "# Shortest Paths\n",
+ "\n",
+ "```{index} single: Dynamic Programming; Shortest Paths\n",
+ "```\n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "The shortest path problem is a [classic problem](https://en.wikipedia.org/wiki/Shortest_path) in mathematics and computer science with applications in\n",
+ "\n",
+ "* Economics (sequential decision making, analysis of social networks, etc.)\n",
+ "* Operations research and transportation\n",
+ "* Robotics and artificial intelligence\n",
+ "* Telecommunication network design and routing\n",
+ "* etc., etc.\n",
+ "\n",
+ "Variations of the methods we discuss in this lecture are used millions of times every day, in applications such as\n",
+ "\n",
+ "* Google Maps\n",
+ "* routing packets on the internet\n",
+ "\n",
+ "For us, the shortest path problem also provides a nice introduction to the logic of **dynamic programming**.\n",
+ "\n",
+ "Dynamic programming is an extremely powerful optimization technique that we apply in many lectures on this site.\n",
+ "\n",
+ "The only scientific library we'll need in what follows is NumPy:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4fc6bf54",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3d491519",
+ "metadata": {},
+ "source": [
+ "## Outline of the problem\n",
+ "\n",
+ "The shortest path problem is one of finding how to traverse a [graph](https://en.wikipedia.org/wiki/Graph_%28mathematics%29) from one specified node to another at minimum cost.\n",
+ "\n",
+ "Consider the following graph\n",
+ "\n",
+ "```{figure} /_static/lecture_specific/short_path/graph.png\n",
+ "\n",
+ "```\n",
+ "\n",
+ "We wish to travel from node (vertex) A to node G at minimum cost\n",
+ "\n",
+ "* Arrows (edges) indicate the movements we can take.\n",
+ "* Numbers on edges indicate the cost of traveling that edge.\n",
+ "\n",
+ "(Graphs such as the one above are called weighted [directed graphs](https://en.wikipedia.org/wiki/Directed_graph).)\n",
+ "\n",
+ "Possible interpretations of the graph include\n",
+ "\n",
+ "* Minimum cost for supplier to reach a destination.\n",
+ "* Routing of packets on the internet (minimize time).\n",
+ "* etc., etc.\n",
+ "\n",
+ "For this simple graph, a quick scan of the edges shows that the optimal paths are\n",
+ "\n",
+ "* A, C, F, G at cost 8\n",
+ "\n",
+ "```{figure} /_static/lecture_specific/short_path/graph4.png\n",
+ "\n",
+ "```\n",
+ "\n",
+ "* A, D, F, G at cost 8\n",
+ "\n",
+ "```{figure} /_static/lecture_specific/short_path/graph3.png\n",
+ "\n",
+ "```\n",
+ "\n",
+ "## Finding least-cost paths\n",
+ "\n",
+ "For large graphs, we need a systematic solution.\n",
+ "\n",
+ "Let $J(v)$ denote the minimum cost-to-go from node $v$, understood as the total cost from $v$ if we take the best route.\n",
+ "\n",
+ "Suppose that we know $J(v)$ for each node $v$, as shown below for the graph from the preceding example.\n",
+ "\n",
+ "```{figure} /_static/lecture_specific/short_path/graph2.png\n",
+ "\n",
+ "```\n",
+ "\n",
+ "Note that $J(G) = 0$.\n",
+ "\n",
+ "The best path can now be found as follows\n",
+ "\n",
+ "1. Start at node $v = A$\n",
+ "1. From current node $v$, move to any node that solves\n",
+ "\n",
+ "```{math}\n",
+ ":label: spprebell\n",
+ "\n",
+ "\\min_{w \\in F_v} \\{ c(v, w) + J(w) \\}\n",
+ "```\n",
+ "\n",
+ "where\n",
+ "\n",
+ "* $F_v$ is the set of nodes that can be reached from $v$ in one step.\n",
+ "* $c(v, w)$ is the cost of traveling from $v$ to $w$.\n",
+ "\n",
+ "Hence, if we know the function $J$, then finding the best path is almost trivial.\n",
+ "\n",
+ "But how can we find the cost-to-go function $J$?\n",
+ "\n",
+ "Some thought will convince you that, for every node $v$,\n",
+ "the function $J$ satisfies\n",
+ "\n",
+ "```{math}\n",
+ ":label: spbell\n",
+ "\n",
+ "J(v) = \\min_{w \\in F_v} \\{ c(v, w) + J(w) \\}\n",
+ "```\n",
+ "\n",
+ "This is known as the **Bellman equation**, after the mathematician [Richard Bellman](https://en.wikipedia.org/wiki/Richard_E._Bellman).\n",
+ "\n",
+ "The Bellman equation can be thought of as a restriction that $J$ must\n",
+ "satisfy.\n",
+ "\n",
+ "What we want to do now is use this restriction to compute $J$.\n",
+ "\n",
+ "## Solving for minimum cost-to-go\n",
+ "\n",
+ "Let's look at an algorithm for computing $J$ and then think about how to\n",
+ "implement it.\n",
+ "\n",
+ "### The algorithm\n",
+ "\n",
+ "The standard algorithm for finding $J$ is to start an initial guess and then iterate.\n",
+ "\n",
+ "This is a standard approach to solving nonlinear equations, often called\n",
+ "the method of **successive approximations**.\n",
+ "\n",
+ "Our initial guess will be\n",
+ "\n",
+ "```{math}\n",
+ ":label: spguess\n",
+ "\n",
+ "J_0(v) = 0 \\text{ for all } v\n",
+ "```\n",
+ "\n",
+ "Now\n",
+ "\n",
+ "1. Set $n = 0$\n",
+ "1. Set $J_{n+1} (v) = \\min_{w \\in F_v} \\{ c(v, w) + J_n(w) \\}$ for all $v$\n",
+ "1. If $J_{n+1}$ and $J_n$ are not equal then increment $n$, go to 2\n",
+ "\n",
+ "This sequence converges to $J$.\n",
+ "\n",
+ "Although we omit the proof, we'll prove similar claims in our other lectures\n",
+ "on dynamic programming.\n",
+ "\n",
+ "### Implementation\n",
+ "\n",
+ "Having an algorithm is a good start, but we also need to think about how to\n",
+ "implement it on a computer.\n",
+ "\n",
+ "First, for the cost function $c$, we'll implement it as a matrix\n",
+ "$Q$, where a typical element is\n",
+ "\n",
+ "$$\n",
+ "Q(v, w)\n",
+ "=\n",
+ "\\begin{cases}\n",
+ " & c(v, w) \\text{ if } w \\in F_v \\\\\n",
+ " & +\\infty \\text{ otherwise }\n",
+ "\\end{cases}\n",
+ "$$\n",
+ "\n",
+ "In this context $Q$ is usually called the **distance matrix**.\n",
+ "\n",
+ "We're also numbering the nodes now, with $A = 0$, so, for example\n",
+ "\n",
+ "$$\n",
+ "Q(1, 2)\n",
+ "=\n",
+ "\\text{ the cost of traveling from B to C }\n",
+ "$$\n",
+ "\n",
+ "For example, for the simple graph above, we set"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b581de5d",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from numpy import inf\n",
+ "\n",
+ "Q = np.array([[inf, 1, 5, 3, inf, inf, inf],\n",
+ " [inf, inf, inf, 9, 6, inf, inf],\n",
+ " [inf, inf, inf, inf, inf, 2, inf],\n",
+ " [inf, inf, inf, inf, inf, 4, 8],\n",
+ " [inf, inf, inf, inf, inf, inf, 4],\n",
+ " [inf, inf, inf, inf, inf, inf, 1],\n",
+ " [inf, inf, inf, inf, inf, inf, 0]])"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cc305e5b",
+ "metadata": {},
+ "source": [
+ "Notice that the cost of staying still (on the principle diagonal) is set to\n",
+ "\n",
+ "* `np.inf` for non-destination nodes --- moving on is required.\n",
+ "* 0 for the destination node --- here is where we stop.\n",
+ "\n",
+ "For the sequence of approximations $\\{J_n\\}$ of the cost-to-go functions, we can use NumPy arrays.\n",
+ "\n",
+ "Let's try with this example and see how we go:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "27589c50",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "nodes = range(7) # Nodes = 0, 1, ..., 6\n",
+ "J = np.zeros_like(nodes, dtype=int) # Initial guess\n",
+ "next_J = np.empty_like(nodes, dtype=int) # Stores updated guess\n",
+ "\n",
+ "max_iter = 500\n",
+ "i = 0\n",
+ "\n",
+ "while i < max_iter:\n",
+ " for v in nodes:\n",
+ " # Minimize Q[v, w] + J[w] over all choices of w\n",
+ " next_J[v] = np.min(Q[v, :] + J)\n",
+ " \n",
+ " if np.array_equal(next_J, J): \n",
+ " break\n",
+ " \n",
+ " J[:] = next_J # Copy contents of next_J to J\n",
+ " i += 1\n",
+ "\n",
+ "print(\"The cost-to-go function is\", J)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "c4ffe4f7",
+ "metadata": {},
+ "source": [
+ "This matches with the numbers we obtained by inspection above.\n",
+ "\n",
+ "But, importantly, we now have a methodology for tackling large graphs.\n",
+ "\n",
+ "## Exercises\n",
+ "\n",
+ "\n",
+ "```{exercise-start}\n",
+ ":label: short_path_ex1\n",
+ "```\n",
+ "\n",
+ "The text below describes a weighted directed graph.\n",
+ "\n",
+ "The line `node0, node1 0.04, node8 11.11, node14 72.21` means that from node0 we can go to\n",
+ "\n",
+ "* node1 at cost 0.04\n",
+ "* node8 at cost 11.11\n",
+ "* node14 at cost 72.21\n",
+ "\n",
+ "No other nodes can be reached directly from node0.\n",
+ "\n",
+ "Other lines have a similar interpretation.\n",
+ "\n",
+ "Your task is to use the algorithm given above to find the optimal path and its cost.\n",
+ "\n",
+ "```{note}\n",
+ "You will be dealing with floating point numbers now, rather than\n",
+ "integers, so consider replacing `np.equal()` with `np.allclose()`.\n",
+ "```"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d454a897",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "%%file graph.txt\n",
+ "node0, node1 0.04, node8 11.11, node14 72.21\n",
+ "node1, node46 1247.25, node6 20.59, node13 64.94\n",
+ "node2, node66 54.18, node31 166.80, node45 1561.45\n",
+ "node3, node20 133.65, node6 2.06, node11 42.43\n",
+ "node4, node75 3706.67, node5 0.73, node7 1.02\n",
+ "node5, node45 1382.97, node7 3.33, node11 34.54\n",
+ "node6, node31 63.17, node9 0.72, node10 13.10\n",
+ "node7, node50 478.14, node9 3.15, node10 5.85\n",
+ "node8, node69 577.91, node11 7.45, node12 3.18\n",
+ "node9, node70 2454.28, node13 4.42, node20 16.53\n",
+ "node10, node89 5352.79, node12 1.87, node16 25.16\n",
+ "node11, node94 4961.32, node18 37.55, node20 65.08\n",
+ "node12, node84 3914.62, node24 34.32, node28 170.04\n",
+ "node13, node60 2135.95, node38 236.33, node40 475.33\n",
+ "node14, node67 1878.96, node16 2.70, node24 38.65\n",
+ "node15, node91 3597.11, node17 1.01, node18 2.57\n",
+ "node16, node36 392.92, node19 3.49, node38 278.71\n",
+ "node17, node76 783.29, node22 24.78, node23 26.45\n",
+ "node18, node91 3363.17, node23 16.23, node28 55.84\n",
+ "node19, node26 20.09, node20 0.24, node28 70.54\n",
+ "node20, node98 3523.33, node24 9.81, node33 145.80\n",
+ "node21, node56 626.04, node28 36.65, node31 27.06\n",
+ "node22, node72 1447.22, node39 136.32, node40 124.22\n",
+ "node23, node52 336.73, node26 2.66, node33 22.37\n",
+ "node24, node66 875.19, node26 1.80, node28 14.25\n",
+ "node25, node70 1343.63, node32 36.58, node35 45.55\n",
+ "node26, node47 135.78, node27 0.01, node42 122.00\n",
+ "node27, node65 480.55, node35 48.10, node43 246.24\n",
+ "node28, node82 2538.18, node34 21.79, node36 15.52\n",
+ "node29, node64 635.52, node32 4.22, node33 12.61\n",
+ "node30, node98 2616.03, node33 5.61, node35 13.95\n",
+ "node31, node98 3350.98, node36 20.44, node44 125.88\n",
+ "node32, node97 2613.92, node34 3.33, node35 1.46\n",
+ "node33, node81 1854.73, node41 3.23, node47 111.54\n",
+ "node34, node73 1075.38, node42 51.52, node48 129.45\n",
+ "node35, node52 17.57, node41 2.09, node50 78.81\n",
+ "node36, node71 1171.60, node54 101.08, node57 260.46\n",
+ "node37, node75 269.97, node38 0.36, node46 80.49\n",
+ "node38, node93 2767.85, node40 1.79, node42 8.78\n",
+ "node39, node50 39.88, node40 0.95, node41 1.34\n",
+ "node40, node75 548.68, node47 28.57, node54 53.46\n",
+ "node41, node53 18.23, node46 0.28, node54 162.24\n",
+ "node42, node59 141.86, node47 10.08, node72 437.49\n",
+ "node43, node98 2984.83, node54 95.06, node60 116.23\n",
+ "node44, node91 807.39, node46 1.56, node47 2.14\n",
+ "node45, node58 79.93, node47 3.68, node49 15.51\n",
+ "node46, node52 22.68, node57 27.50, node67 65.48\n",
+ "node47, node50 2.82, node56 49.31, node61 172.64\n",
+ "node48, node99 2564.12, node59 34.52, node60 66.44\n",
+ "node49, node78 53.79, node50 0.51, node56 10.89\n",
+ "node50, node85 251.76, node53 1.38, node55 20.10\n",
+ "node51, node98 2110.67, node59 23.67, node60 73.79\n",
+ "node52, node94 1471.80, node64 102.41, node66 123.03\n",
+ "node53, node72 22.85, node56 4.33, node67 88.35\n",
+ "node54, node88 967.59, node59 24.30, node73 238.61\n",
+ "node55, node84 86.09, node57 2.13, node64 60.80\n",
+ "node56, node76 197.03, node57 0.02, node61 11.06\n",
+ "node57, node86 701.09, node58 0.46, node60 7.01\n",
+ "node58, node83 556.70, node64 29.85, node65 34.32\n",
+ "node59, node90 820.66, node60 0.72, node71 0.67\n",
+ "node60, node76 48.03, node65 4.76, node67 1.63\n",
+ "node61, node98 1057.59, node63 0.95, node64 4.88\n",
+ "node62, node91 132.23, node64 2.94, node76 38.43\n",
+ "node63, node66 4.43, node72 70.08, node75 56.34\n",
+ "node64, node80 47.73, node65 0.30, node76 11.98\n",
+ "node65, node94 594.93, node66 0.64, node73 33.23\n",
+ "node66, node98 395.63, node68 2.66, node73 37.53\n",
+ "node67, node82 153.53, node68 0.09, node70 0.98\n",
+ "node68, node94 232.10, node70 3.35, node71 1.66\n",
+ "node69, node99 247.80, node70 0.06, node73 8.99\n",
+ "node70, node76 27.18, node72 1.50, node73 8.37\n",
+ "node71, node89 104.50, node74 8.86, node91 284.64\n",
+ "node72, node76 15.32, node84 102.77, node92 133.06\n",
+ "node73, node83 52.22, node76 1.40, node90 243.00\n",
+ "node74, node81 1.07, node76 0.52, node78 8.08\n",
+ "node75, node92 68.53, node76 0.81, node77 1.19\n",
+ "node76, node85 13.18, node77 0.45, node78 2.36\n",
+ "node77, node80 8.94, node78 0.98, node86 64.32\n",
+ "node78, node98 355.90, node81 2.59\n",
+ "node79, node81 0.09, node85 1.45, node91 22.35\n",
+ "node80, node92 121.87, node88 28.78, node98 264.34\n",
+ "node81, node94 99.78, node89 39.52, node92 99.89\n",
+ "node82, node91 47.44, node88 28.05, node93 11.99\n",
+ "node83, node94 114.95, node86 8.75, node88 5.78\n",
+ "node84, node89 19.14, node94 30.41, node98 121.05\n",
+ "node85, node97 94.51, node87 2.66, node89 4.90\n",
+ "node86, node97 85.09\n",
+ "node87, node88 0.21, node91 11.14, node92 21.23\n",
+ "node88, node93 1.31, node91 6.83, node98 6.12\n",
+ "node89, node97 36.97, node99 82.12\n",
+ "node90, node96 23.53, node94 10.47, node99 50.99\n",
+ "node91, node97 22.17\n",
+ "node92, node96 10.83, node97 11.24, node99 34.68\n",
+ "node93, node94 0.19, node97 6.71, node99 32.77\n",
+ "node94, node98 5.91, node96 2.03\n",
+ "node95, node98 6.17, node99 0.27\n",
+ "node96, node98 3.32, node97 0.43, node99 5.87\n",
+ "node97, node98 0.30\n",
+ "node98, node99 0.33\n",
+ "node99,"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6e516954",
+ "metadata": {},
+ "source": [
+ "```{exercise-end}\n",
+ "```\n",
+ "\n",
+ "```{solution-start} short_path_ex1\n",
+ ":class: dropdown\n",
+ "```\n",
+ "\n",
+ "First let's write a function that reads in the graph data above and builds a distance matrix."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3094573a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "num_nodes = 100\n",
+ "destination_node = 99\n",
+ "\n",
+ "def map_graph_to_distance_matrix(in_file):\n",
+ "\n",
+ " # First let's set of the distance matrix Q with inf everywhere\n",
+ " Q = np.full((num_nodes, num_nodes), np.inf)\n",
+ "\n",
+ " # Now we read in the data and modify Q\n",
+ " with open(in_file) as infile:\n",
+ " for line in infile:\n",
+ " elements = line.split(',')\n",
+ " node = elements.pop(0)\n",
+ " node = int(node[4:]) # convert node description to integer\n",
+ " if node != destination_node:\n",
+ " for element in elements:\n",
+ " destination, cost = element.split()\n",
+ " destination = int(destination[4:])\n",
+ " Q[node, destination] = float(cost)\n",
+ " Q[destination_node, destination_node] = 0\n",
+ " return Q"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "56e5487a",
+ "metadata": {},
+ "source": [
+ "In addition, let's write\n",
+ "\n",
+ "1. a \"Bellman operator\" function that takes a distance matrix and current guess of J and returns an updated guess of J, and\n",
+ "1. a function that takes a distance matrix and returns a cost-to-go function.\n",
+ "\n",
+ "We'll use the algorithm described above.\n",
+ "\n",
+ "The minimization step is vectorized to make it faster."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "5baa2d4e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def bellman(J, Q):\n",
+ " return np.min(Q + J, axis=1)\n",
+ "\n",
+ "\n",
+ "def compute_cost_to_go(Q):\n",
+ " num_nodes = Q.shape[0]\n",
+ " J = np.zeros(num_nodes) # Initial guess\n",
+ " max_iter = 500\n",
+ " i = 0\n",
+ "\n",
+ " while i < max_iter:\n",
+ " next_J = bellman(J, Q)\n",
+ " if np.allclose(next_J, J):\n",
+ " break\n",
+ " else:\n",
+ " J[:] = next_J # Copy contents of next_J to J\n",
+ " i += 1\n",
+ "\n",
+ " return(J)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "af4c1334",
+ "metadata": {},
+ "source": [
+ "We used np.allclose() rather than testing exact equality because we are\n",
+ "dealing with floating point numbers now.\n",
+ "\n",
+ "Finally, here's a function that uses the cost-to-go function to obtain the\n",
+ "optimal path (and its cost)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "25ce9e7f",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def print_best_path(J, Q):\n",
+ " sum_costs = 0\n",
+ " current_node = 0\n",
+ " while current_node != destination_node:\n",
+ " print(current_node)\n",
+ " # Move to the next node and increment costs\n",
+ " next_node = np.argmin(Q[current_node, :] + J)\n",
+ " sum_costs += Q[current_node, next_node]\n",
+ " current_node = next_node\n",
+ "\n",
+ " print(destination_node)\n",
+ " print('Cost: ', sum_costs)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "db899e6a",
+ "metadata": {},
+ "source": [
+ "Okay, now we have the necessary functions, let's call them to do the job we were assigned."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9dc5d1cc",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "Q = map_graph_to_distance_matrix('graph.txt')\n",
+ "J = compute_cost_to_go(Q)\n",
+ "print_best_path(J, Q)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "482c6e46",
+ "metadata": {},
+ "source": [
+ "The total cost of the path should agree with $J[0]$ so let's check this."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "19a0bf59",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "J[0]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d6090662",
+ "metadata": {},
+ "source": [
+ "```{solution-end}\n",
+ "```"
+ ]
+ }
+ ],
+ "metadata": {
+ "jupytext": {
+ "text_representation": {
+ "extension": ".md",
+ "format_name": "myst"
+ }
+ },
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "source_map": [
+ 10,
+ 47,
+ 49,
+ 198,
+ 208,
+ 219,
+ 239,
+ 271,
+ 373,
+ 384,
+ 406,
+ 417,
+ 437,
+ 445,
+ 458,
+ 462,
+ 466,
+ 470,
+ 472
+ ]
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/lectures/short_path.md b/_sources/short_path.md
similarity index 100%
rename from lectures/short_path.md
rename to _sources/short_path.md
diff --git a/_sources/simple_linear_regression.ipynb b/_sources/simple_linear_regression.ipynb
new file mode 100644
index 000000000..0ef886b73
--- /dev/null
+++ b/_sources/simple_linear_regression.ipynb
@@ -0,0 +1,1086 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "830004c2",
+ "metadata": {},
+ "source": [
+ "# Simple Linear Regression Model"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "81dd8f10",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import pandas as pd\n",
+ "import matplotlib.pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1609cdd1",
+ "metadata": {},
+ "source": [
+ "The simple regression model estimates the relationship between two variables $x_i$ and $y_i$\n",
+ "\n",
+ "$$\n",
+ "y_i = \\alpha + \\beta x_i + \\epsilon_i, i = 1,2,...,N\n",
+ "$$\n",
+ "\n",
+ "where $\\epsilon_i$ represents the error between the line of best fit and the sample values for $y_i$ given $x_i$.\n",
+ "\n",
+ "Our goal is to choose values for $\\alpha$ and $\\beta$ to build a line of \"best\" fit for some data that is available for variables $x_i$ and $y_i$. \n",
+ "\n",
+ "Let us consider a simple dataset of 10 observations for variables $x_i$ and $y_i$:\n",
+ "\n",
+ "| | $y_i$ | $x_i$ |\n",
+ "|-|---|---|\n",
+ "|1| 2000 | 32 |\n",
+ "|2| 1000 | 21 | \n",
+ "|3| 1500 | 24 | \n",
+ "|4| 2500 | 35 | \n",
+ "|5| 500 | 10 |\n",
+ "|6| 900 | 11 |\n",
+ "|7| 1100 | 22 | \n",
+ "|8| 1500 | 21 | \n",
+ "|9| 1800 | 27 |\n",
+ "|10 | 250 | 2 |\n",
+ "\n",
+ "Let us think about $y_i$ as sales for an ice-cream cart, while $x_i$ is a variable that records the day's temperature in Celsius."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b159bef3",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "x = [32, 21, 24, 35, 10, 11, 22, 21, 27, 2]\n",
+ "y = [2000,1000,1500,2500,500,900,1100,1500,1800, 250]\n",
+ "df = pd.DataFrame([x,y]).T\n",
+ "df.columns = ['X', 'Y']\n",
+ "df"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "3fa57c60",
+ "metadata": {},
+ "source": [
+ "We can use a scatter plot of the data to see the relationship between $y_i$ (ice-cream sales in dollars (\\$\\'s)) and $x_i$ (degrees Celsius)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d5619a98",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Scatter plot",
+ "name": "sales-v-temp1"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "ax = df.plot(\n",
+ " x='X', \n",
+ " y='Y', \n",
+ " kind='scatter', \n",
+ " ylabel='Ice-cream sales ($\\'s)', \n",
+ " xlabel='Degrees celcius'\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4de3a837",
+ "metadata": {},
+ "source": [
+ "as you can see the data suggests that more ice-cream is typically sold on hotter days. \n",
+ "\n",
+ "To build a linear model of the data we need to choose values for $\\alpha$ and $\\beta$ that represents a line of \"best\" fit such that\n",
+ "\n",
+ "$$\n",
+ "\\hat{y_i} = \\hat{\\alpha} + \\hat{\\beta} x_i\n",
+ "$$\n",
+ "\n",
+ "Let's start with $\\alpha = 5$ and $\\beta = 10$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "c447a8fb",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "α = 5\n",
+ "β = 10\n",
+ "df['Y_hat'] = α + β * df['X']"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "e880629f",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Scatter plot with a line of fit",
+ "name": "sales-v-temp2"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax = df.plot(x='X',y='Y', kind='scatter', ax=ax)\n",
+ "ax = df.plot(x='X',y='Y_hat', kind='line', ax=ax)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "795e01a7",
+ "metadata": {},
+ "source": [
+ "We can see that this model does a poor job of estimating the relationship.\n",
+ "\n",
+ "We can continue to guess and iterate towards a line of \"best\" fit by adjusting the parameters"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1cdc6355",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "β = 100\n",
+ "df['Y_hat'] = α + β * df['X']"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "0c2b3f44",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Scatter plot with a line of fit #2",
+ "name": "sales-v-temp3"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax = df.plot(x='X',y='Y', kind='scatter', ax=ax)\n",
+ "ax = df.plot(x='X',y='Y_hat', kind='line', ax=ax)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "88f90f4f",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "β = 65\n",
+ "df['Y_hat'] = α + β * df['X']"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "7544f695",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Scatter plot with a line of fit #3",
+ "name": "sales-v-temp4"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax = df.plot(x='X',y='Y', kind='scatter', ax=ax)\n",
+ "ax = df.plot(x='X',y='Y_hat', kind='line', ax=ax, color='g')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "9fc07442",
+ "metadata": {},
+ "source": [
+ "However we need to think about formalizing this guessing process by thinking of this problem as an optimization problem. \n",
+ "\n",
+ "Let's consider the error $\\epsilon_i$ and define the difference between the observed values $y_i$ and the estimated values $\\hat{y}_i$ which we will call the residuals\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "\\hat{e}_i &= y_i - \\hat{y}_i \\\\\n",
+ " &= y_i - \\hat{\\alpha} - \\hat{\\beta} x_i\n",
+ "\\end{aligned}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4a75a1ca",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "df['error'] = df['Y_hat'] - df['Y']"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "4580282f",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "df"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "95aee3a8",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Plot of the residuals",
+ "name": "plt-residuals"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "fig, ax = plt.subplots()\n",
+ "ax = df.plot(x='X',y='Y', kind='scatter', ax=ax)\n",
+ "ax = df.plot(x='X',y='Y_hat', kind='line', ax=ax, color='g')\n",
+ "plt.vlines(df['X'], df['Y_hat'], df['Y'], color='r')\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5356f771",
+ "metadata": {},
+ "source": [
+ "The Ordinary Least Squares (OLS) method chooses $\\alpha$ and $\\beta$ in such a way that **minimizes** the sum of the squared residuals (SSR). \n",
+ "\n",
+ "$$\n",
+ "\\min_{\\alpha,\\beta} \\sum_{i=1}^{N}{\\hat{e}_i^2} = \\min_{\\alpha,\\beta} \\sum_{i=1}^{N}{(y_i - \\alpha - \\beta x_i)^2}\n",
+ "$$\n",
+ "\n",
+ "Let's call this a cost function\n",
+ "\n",
+ "$$\n",
+ "C = \\sum_{i=1}^{N}{(y_i - \\alpha - \\beta x_i)^2}\n",
+ "$$\n",
+ "\n",
+ "that we would like to minimize with parameters $\\alpha$ and $\\beta$.\n",
+ "\n",
+ "## How does error change with respect to $\\alpha$ and $\\beta$\n",
+ "\n",
+ "Let us first look at how the total error changes with respect to $\\beta$ (holding the intercept $\\alpha$ constant)\n",
+ "\n",
+ "We know from [the next section](slr:optimal-values) the optimal values for $\\alpha$ and $\\beta$ are:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "38ba7927",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "β_optimal = 64.38\n",
+ "α_optimal = -14.72"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5f10af1c",
+ "metadata": {},
+ "source": [
+ "We can then calculate the error for a range of $\\beta$ values"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9fcb1536",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "errors = {}\n",
+ "for β in np.arange(20,100,0.5):\n",
+ " errors[β] = abs((α_optimal + β * df['X']) - df['Y']).sum()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a73ac67b",
+ "metadata": {},
+ "source": [
+ "Plotting the error"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3d2a2d34",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Plotting the error",
+ "name": "plt-errors"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "ax = pd.Series(errors).plot(xlabel='β', ylabel='error')\n",
+ "plt.axvline(β_optimal, color='r');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6b55a2b9",
+ "metadata": {},
+ "source": [
+ "Now let us vary $\\alpha$ (holding $\\beta$ constant)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a46c18a4",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "errors = {}\n",
+ "for α in np.arange(-500,500,5):\n",
+ " errors[α] = abs((α + β_optimal * df['X']) - df['Y']).sum()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "935104c9",
+ "metadata": {},
+ "source": [
+ "Plotting the error"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "35b93c66",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Plotting the error (2)",
+ "name": "plt-errors-2"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "ax = pd.Series(errors).plot(xlabel='α', ylabel='error')\n",
+ "plt.axvline(α_optimal, color='r');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5a7cc895",
+ "metadata": {},
+ "source": [
+ "(slr:optimal-values)=\n",
+ "## Calculating optimal values\n",
+ "\n",
+ "Now let us use calculus to solve the optimization problem and compute the optimal values for $\\alpha$ and $\\beta$ to find the ordinary least squares solution.\n",
+ "\n",
+ "First taking the partial derivative with respect to $\\alpha$\n",
+ "\n",
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\alpha}[\\sum_{i=1}^{N}{(y_i - \\alpha - \\beta x_i)^2}]\n",
+ "$$\n",
+ "\n",
+ "and setting it equal to $0$\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{-2(y_i - \\alpha - \\beta x_i)}\n",
+ "$$\n",
+ "\n",
+ "we can remove the constant $-2$ from the summation by dividing both sides by $-2$\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{(y_i - \\alpha - \\beta x_i)}\n",
+ "$$\n",
+ "\n",
+ "Now we can split this equation up into the components\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{y_i} - \\sum_{i=1}^{N}{\\alpha} - \\beta \\sum_{i=1}^{N}{x_i}\n",
+ "$$\n",
+ "\n",
+ "The middle term is a straight forward sum from $i=1,...N$ by a constant $\\alpha$\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{y_i} - N*\\alpha - \\beta \\sum_{i=1}^{N}{x_i}\n",
+ "$$\n",
+ "\n",
+ "and rearranging terms \n",
+ "\n",
+ "$$\n",
+ "\\alpha = \\frac{\\sum_{i=1}^{N}{y_i} - \\beta \\sum_{i=1}^{N}{x_i}}{N}\n",
+ "$$\n",
+ "\n",
+ "We observe that both fractions resolve to the means $\\bar{y_i}$ and $\\bar{x_i}$\n",
+ "\n",
+ "$$\n",
+ "\\alpha = \\bar{y_i} - \\beta\\bar{x_i}\n",
+ "$$ (eq:optimal-alpha)\n",
+ "\n",
+ "Now let's take the partial derivative of the cost function $C$ with respect to $\\beta$\n",
+ "\n",
+ "$$\n",
+ "\\frac{\\partial C}{\\partial \\beta}[\\sum_{i=1}^{N}{(y_i - \\alpha - \\beta x_i)^2}]\n",
+ "$$\n",
+ "\n",
+ "and setting it equal to $0$\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{-2 x_i (y_i - \\alpha - \\beta x_i)}\n",
+ "$$\n",
+ "\n",
+ "we can again take the constant outside of the summation and divide both sides by $-2$\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{x_i (y_i - \\alpha - \\beta x_i)}\n",
+ "$$\n",
+ "\n",
+ "which becomes\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{(x_i y_i - \\alpha x_i - \\beta x_i^2)}\n",
+ "$$\n",
+ "\n",
+ "now substituting for $\\alpha$\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{(x_i y_i - (\\bar{y_i} - \\beta \\bar{x_i}) x_i - \\beta x_i^2)}\n",
+ "$$\n",
+ "\n",
+ "and rearranging terms\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}{(x_i y_i - \\bar{y_i} x_i - \\beta \\bar{x_i} x_i - \\beta x_i^2)}\n",
+ "$$\n",
+ "\n",
+ "This can be split into two summations\n",
+ "\n",
+ "$$\n",
+ "0 = \\sum_{i=1}^{N}(x_i y_i - \\bar{y_i} x_i) + \\beta \\sum_{i=1}^{N}(\\bar{x_i} x_i - x_i^2)\n",
+ "$$\n",
+ "\n",
+ "and solving for $\\beta$ yields\n",
+ "\n",
+ "$$\n",
+ "\\beta = \\frac{\\sum_{i=1}^{N}(x_i y_i - \\bar{y_i} x_i)}{\\sum_{i=1}^{N}(x_i^2 - \\bar{x_i} x_i)}\n",
+ "$$ (eq:optimal-beta)\n",
+ "\n",
+ "We can now use {eq}`eq:optimal-alpha` and {eq}`eq:optimal-beta` to calculate the optimal values for $\\alpha$ and $\\beta$\n",
+ "\n",
+ "Calculating $\\beta$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "86767dfa",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "df = df[['X','Y']].copy() # Original Data\n",
+ "\n",
+ "# Calculate the sample means\n",
+ "x_bar = df['X'].mean()\n",
+ "y_bar = df['Y'].mean()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bd140f18",
+ "metadata": {},
+ "source": [
+ "Now computing across the 10 observations and then summing the numerator and denominator"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8cfaf446",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Compute the Sums\n",
+ "df['num'] = df['X'] * df['Y'] - y_bar * df['X']\n",
+ "df['den'] = pow(df['X'],2) - x_bar * df['X']\n",
+ "β = df['num'].sum() / df['den'].sum()\n",
+ "print(β)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7adf3ed2",
+ "metadata": {},
+ "source": [
+ "Calculating $\\alpha$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f7d6bc01",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "α = y_bar - β * x_bar\n",
+ "print(α)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "148c46f9",
+ "metadata": {},
+ "source": [
+ "Now we can plot the OLS solution"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "3d7de6d7",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "OLS line of best fit",
+ "name": "plt-ols"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "df['Y_hat'] = α + β * df['X']\n",
+ "df['error'] = df['Y_hat'] - df['Y']\n",
+ "\n",
+ "fig, ax = plt.subplots()\n",
+ "ax = df.plot(x='X',y='Y', kind='scatter', ax=ax)\n",
+ "ax = df.plot(x='X',y='Y_hat', kind='line', ax=ax, color='g')\n",
+ "plt.vlines(df['X'], df['Y_hat'], df['Y'], color='r');"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "b1aaad94",
+ "metadata": {},
+ "source": [
+ ":::{exercise}\n",
+ ":label: slr-ex1\n",
+ "\n",
+ "Now that you know the equations that solve the simple linear regression model using OLS you can now run your own regressions to build a model between $y$ and $x$.\n",
+ "\n",
+ "Let's consider two economic variables GDP per capita and Life Expectancy.\n",
+ "\n",
+ "1. What do you think their relationship would be?\n",
+ "2. Gather some data [from our world in data](https://ourworldindata.org)\n",
+ "3. Use `pandas` to import the `csv` formatted data and plot a few different countries of interest\n",
+ "4. Use {eq}`eq:optimal-alpha` and {eq}`eq:optimal-beta` to compute optimal values for $\\alpha$ and $\\beta$\n",
+ "5. Plot the line of best fit found using OLS\n",
+ "6. Interpret the coefficients and write a summary sentence of the relationship between GDP per capita and Life Expectancy\n",
+ "\n",
+ ":::\n",
+ "\n",
+ ":::{solution-start} slr-ex1\n",
+ ":::\n",
+ "\n",
+ "**Q2:** Gather some data [from our world in data](https://ourworldindata.org)\n",
+ "\n",
+ ":::{raw} html\n",
+ "\n",
+ ":::\n",
+ "\n",
+ "You can download {download}`a copy of the data here \n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "# Univariate Time Series with Matrix Algebra\n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "This lecture uses matrices to solve some linear difference equations.\n",
+ "\n",
+ "As a running example, we’ll study a **second-order linear difference\n",
+ "equation** that was the key technical tool in Paul Samuelson’s 1939\n",
+ "article {cite}`Samuelson1939` that introduced the *multiplier-accelerator model*.\n",
+ "\n",
+ "This model became the workhorse that powered early econometric versions of\n",
+ "Keynesian macroeconomic models in the United States.\n",
+ "\n",
+ "You can read about the details of that model in {doc}`intermediate:samuelson`.\n",
+ "\n",
+ "(That lecture also describes some technicalities about second-order linear difference equations.)\n",
+ "\n",
+ "In this lecture, we'll also learn about an **autoregressive** representation and a **moving average** representation of a non-stationary\n",
+ "univariate time series $\\{y_t\\}_{t=0}^T$.\n",
+ "\n",
+ "We'll also study a \"perfect foresight\" model of stock prices that involves solving\n",
+ "a \"forward-looking\" linear difference equation.\n",
+ "\n",
+ "We will use the following imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b2879b17",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from matplotlib import cm\n",
+ "\n",
+ "# Custom figsize for this lecture\n",
+ "plt.rcParams[\"figure.figsize\"] = (11, 5)\n",
+ "\n",
+ "# Set decimal printing to 3 decimal places\n",
+ "np.set_printoptions(precision=3, suppress=True)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "17c588ba",
+ "metadata": {},
+ "source": [
+ "## Samuelson's model\n",
+ "\n",
+ "Let $t = 0, \\pm 1, \\pm 2, \\ldots$ index time.\n",
+ "\n",
+ "For $t = 1, 2, 3, \\ldots, T$ suppose that\n",
+ "\n",
+ "```{math}\n",
+ ":label: tswm_1\n",
+ "\n",
+ "y_{t} = \\alpha_{0} + \\alpha_{1} y_{t-1} + \\alpha_{2} y_{t-2}\n",
+ "```\n",
+ "\n",
+ "where we assume that $y_0$ and $y_{-1}$ are given numbers\n",
+ "that we take as *initial conditions*.\n",
+ "\n",
+ "In Samuelson's model, $y_t$ stood for **national income** or perhaps a different\n",
+ "measure of aggregate activity called **gross domestic product** (GDP) at time $t$.\n",
+ "\n",
+ "Equation {eq}`tswm_1` is called a *second-order linear difference equation*. It is called second order because it depends on two lags.\n",
+ "\n",
+ "But actually, it is a collection of $T$ simultaneous linear\n",
+ "equations in the $T$ variables $y_1, y_2, \\ldots, y_T$.\n",
+ "\n",
+ "```{note}\n",
+ "To be able to solve a second-order linear difference\n",
+ "equation, we require two *boundary conditions* that can take the form\n",
+ "either of two *initial conditions*, two *terminal conditions* or\n",
+ "possibly one of each.\n",
+ "```\n",
+ "\n",
+ "Let’s write our equations as a stacked system\n",
+ "\n",
+ "$$\n",
+ "\\underset{\\equiv A}{\\underbrace{\\left[\\begin{array}{cccccccc}\n",
+ "1 & 0 & 0 & 0 & \\cdots & 0 & 0 & 0\\\\\n",
+ "-\\alpha_{1} & 1 & 0 & 0 & \\cdots & 0 & 0 & 0\\\\\n",
+ "-\\alpha_{2} & -\\alpha_{1} & 1 & 0 & \\cdots & 0 & 0 & 0\\\\\n",
+ "0 & -\\alpha_{2} & -\\alpha_{1} & 1 & \\cdots & 0 & 0 & 0\\\\\n",
+ "\\vdots & \\vdots & \\vdots & \\vdots & \\cdots & \\vdots & \\vdots & \\vdots\\\\\n",
+ "0 & 0 & 0 & 0 & \\cdots & -\\alpha_{2} & -\\alpha_{1} & 1\n",
+ "\\end{array}\\right]}}\\left[\\begin{array}{c}\n",
+ "y_{1}\\\\\n",
+ "y_{2}\\\\\n",
+ "y_{3}\\\\\n",
+ "y_{4}\\\\\n",
+ "\\vdots\\\\\n",
+ "y_{T}\n",
+ "\\end{array}\\right]=\\underset{\\equiv b}{\\underbrace{\\left[\\begin{array}{c}\n",
+ "\\alpha_{0}+\\alpha_{1}y_{0}+\\alpha_{2}y_{-1}\\\\\n",
+ "\\alpha_{0}+\\alpha_{2}y_{0}\\\\\n",
+ "\\alpha_{0}\\\\\n",
+ "\\alpha_{0}\\\\\n",
+ "\\vdots\\\\\n",
+ "\\alpha_{0}\n",
+ "\\end{array}\\right]}}\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "A y = b\n",
+ "$$\n",
+ "\n",
+ "where\n",
+ "\n",
+ "$$\n",
+ "y = \\begin{bmatrix} y_1 \\cr y_2 \\cr \\vdots \\cr y_T \\end{bmatrix}\n",
+ "$$\n",
+ "\n",
+ "Evidently $y$ can be computed from\n",
+ "\n",
+ "$$\n",
+ "y = A^{-1} b\n",
+ "$$\n",
+ "\n",
+ "The vector $y$ is a complete time path $\\{y_t\\}_{t=1}^T$.\n",
+ "\n",
+ "Let’s put Python to work on an example that captures the flavor of\n",
+ "Samuelson’s multiplier-accelerator model.\n",
+ "\n",
+ "We'll set parameters equal to the same values we used in {doc}`intermediate:samuelson`."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ee6e2329",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "T = 80\n",
+ "\n",
+ "# parameters\n",
+ "α_0 = 10.0\n",
+ "α_1 = 1.53\n",
+ "α_2 = -.9\n",
+ "\n",
+ "y_neg1 = 28.0 # y_{-1}\n",
+ "y_0 = 24.0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "387c2e92",
+ "metadata": {},
+ "source": [
+ "Now we construct $A$ and $b$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "33af009a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A = np.identity(T) # The T x T identity matrix\n",
+ "\n",
+ "for i in range(T):\n",
+ "\n",
+ " if i-1 >= 0:\n",
+ " A[i, i-1] = -α_1\n",
+ "\n",
+ " if i-2 >= 0:\n",
+ " A[i, i-2] = -α_2\n",
+ "\n",
+ "b = np.full(T, α_0)\n",
+ "b[0] = α_0 + α_1 * y_0 + α_2 * y_neg1\n",
+ "b[1] = α_0 + α_2 * y_0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "8e290041",
+ "metadata": {},
+ "source": [
+ "Let’s look at the matrix $A$ and the vector $b$ for our\n",
+ "example."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9ee05704",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A, b"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "06a0c83d",
+ "metadata": {},
+ "source": [
+ "Now let’s solve for the path of $y$.\n",
+ "\n",
+ "If $y_t$ is GNP at time $t$, then we have a version of\n",
+ "Samuelson’s model of the dynamics for GNP.\n",
+ "\n",
+ "To solve $y = A^{-1} b$ we can either invert $A$ directly, as in"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d3da1f59",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "A_inv = np.linalg.inv(A)\n",
+ "\n",
+ "y = A_inv @ b"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "bf417887",
+ "metadata": {},
+ "source": [
+ "or we can use `np.linalg.solve`:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "56889940",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "y_second_method = np.linalg.solve(A, b)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5a48660c",
+ "metadata": {},
+ "source": [
+ "Here make sure the two methods give the same result, at least up to floating\n",
+ "point precision:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "860d3d31",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "np.allclose(y, y_second_method)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "6e78e117",
+ "metadata": {},
+ "source": [
+ "$A$ is invertible as it is lower triangular and [its diagonal entries are non-zero](https://www.statlect.com/matrix-algebra/triangular-matrix)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8946e9d3",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Check if A is lower triangular\n",
+ "np.allclose(A, np.tril(A))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "f0ddd3cc",
+ "metadata": {},
+ "source": [
+ "```{note}\n",
+ "In general, `np.linalg.solve` is more numerically stable than using\n",
+ "`np.linalg.inv` directly. \n",
+ "However, stability is not an issue for this small example. Moreover, we will\n",
+ "repeatedly use `A_inv` in what follows, so there is added value in computing\n",
+ "it directly.\n",
+ "```\n",
+ "\n",
+ "Now we can plot."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "9c746c28",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.plot(np.arange(T)+1, y)\n",
+ "plt.xlabel('t')\n",
+ "plt.ylabel('y')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "7e706def",
+ "metadata": {},
+ "source": [
+ "The {ref}`*steady state*\n",
+ " \n",
+ "\n",
+ " \n",
+ " \n",
+ " \n",
+ "
\n",
+ "```\n",
+ "\n",
+ "# Troubleshooting\n",
+ "\n",
+ "This page is for readers experiencing errors when running the code from the lectures.\n",
+ "\n",
+ "## Fixing your local environment\n",
+ "\n",
+ "The basic assumption of the lectures is that code in a lecture should execute whenever\n",
+ "\n",
+ "1. it is executed in a Jupyter notebook and\n",
+ "1. the notebook is running on a machine with the latest version of Anaconda Python.\n",
+ "\n",
+ "You have installed Anaconda, haven't you, following the instructions in [this lecture](https://python-programming.quantecon.org/getting_started.html)?\n",
+ "\n",
+ "Assuming that you have, the most common source of problems for our readers is that their Anaconda distribution is not up to date.\n",
+ "\n",
+ "[Here's a useful article](https://www.anaconda.com/blog/keeping-anaconda-date)\n",
+ "on how to update Anaconda.\n",
+ "\n",
+ "Another option is to simply remove Anaconda and reinstall.\n",
+ "\n",
+ "You also need to keep the external code libraries, such as [QuantEcon.py](https://quantecon.org/quantecon-py) up to date.\n",
+ "\n",
+ "For this task you can either\n",
+ "\n",
+ "* use conda install -y quantecon on the command line, or\n",
+ "* execute !conda install -y quantecon within a Jupyter notebook.\n",
+ "\n",
+ "If your local environment is still not working you can do two things.\n",
+ "\n",
+ "First, you can use a remote machine instead, by clicking on the Launch Notebook icon available for each lecture\n",
+ "\n",
+ "```{image} _static/lecture_specific/troubleshooting/launch.png\n",
+ "\n",
+ "```\n",
+ "\n",
+ "Second, you can report an issue, so we can try to fix your local set up.\n",
+ "\n",
+ "We like getting feedback on the lectures so please don't hesitate to get in\n",
+ "touch.\n",
+ "\n",
+ "## Reporting an issue\n",
+ "\n",
+ "One way to give feedback is to raise an issue through our [issue tracker](https://github.com/QuantEcon/lecture-python/issues).\n",
+ "\n",
+ "Please be as specific as possible. Tell us where the problem is and as much\n",
+ "detail about your local set up as you can provide.\n",
+ "\n",
+ "Another feedback option is to use our [discourse forum](https://discourse.quantecon.org/).\n",
+ "\n",
+ "Finally, you can provide direct feedback to [contact@quantecon.org](mailto:contact@quantecon.org)"
+ ]
+ }
+ ],
+ "metadata": {
+ "jupytext": {
+ "text_representation": {
+ "extension": ".md",
+ "format_name": "myst"
+ }
+ },
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "source_map": [
+ 10
+ ]
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/lectures/troubleshooting.md b/_sources/troubleshooting.md
similarity index 96%
rename from lectures/troubleshooting.md
rename to _sources/troubleshooting.md
index 462301d83..7bc907add 100644
--- a/lectures/troubleshooting.md
+++ b/_sources/troubleshooting.md
@@ -65,5 +65,7 @@ One way to give feedback is to raise an issue through our [issue tracker](https:
Please be as specific as possible. Tell us where the problem is and as much
detail about your local set up as you can provide.
+Another feedback option is to use our [discourse forum](https://discourse.quantecon.org/).
+
Finally, you can provide direct feedback to [contact@quantecon.org](mailto:contact@quantecon.org)
diff --git a/_sources/unpleasant.ipynb b/_sources/unpleasant.ipynb
new file mode 100644
index 000000000..f56e98bc8
--- /dev/null
+++ b/_sources/unpleasant.ipynb
@@ -0,0 +1,622 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "23d200b6",
+ "metadata": {},
+ "source": [
+ "# Some Unpleasant Monetarist Arithmetic \n",
+ "\n",
+ "## Overview\n",
+ "\n",
+ "\n",
+ "This lecture builds on concepts and issues introduced in {doc}`money_inflation`.\n",
+ "\n",
+ "That lecture describes stationary equilibria that reveal a [*Laffer curve*](https://en.wikipedia.org/wiki/Laffer_curve) in the inflation tax rate and the associated stationary rate of return \n",
+ "on currency. \n",
+ "\n",
+ "In this lecture we study a situation in which a stationary equilibrium prevails after date $T > 0$, but not before then. \n",
+ "\n",
+ "For $t=0, \\ldots, T-1$, the money supply, price level, and interest-bearing government debt vary along a transition path that ends at $t=T$.\n",
+ "\n",
+ "During this transition, the ratio of the real balances $\\frac{m_{t+1}}{{p_t}}$ to indexed one-period government bonds $\\tilde R B_{t-1}$ maturing at time $t$ decreases each period. \n",
+ "\n",
+ "This has consequences for the **gross-of-interest** government deficit that must be financed by printing money for times $t \\geq T$. \n",
+ "\n",
+ "The critical **money-to-bonds** ratio stabilizes only at time $T$ and afterwards.\n",
+ "\n",
+ "And the larger is $T$, the higher is the gross-of-interest government deficit that must be financed\n",
+ "by printing money at times $t \\geq T$. \n",
+ "\n",
+ "These outcomes are the essential finding of Sargent and Wallace's \"unpleasant monetarist arithmetic\" {cite}`sargent1981`.\n",
+ "\n",
+ "That lecture described supplies and demands for money that appear in lecture.\n",
+ "\n",
+ "It also characterized the steady state equilibrium from which we work backwards in this lecture. \n",
+ "\n",
+ "In addition to learning about \"unpleasant monetarist arithmetic\", in this lecture we'll learn how to implement a [*fixed point*](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) algorithm for computing an initial price level.\n",
+ "\n",
+ "\n",
+ "## Setup\n",
+ "\n",
+ "Let's start with quick reminders of the model's components set out in {doc}`money_inflation`.\n",
+ "\n",
+ "Please consult that lecture for more details and Python code that we'll also use in this lecture.\n",
+ "\n",
+ "For $t \\geq 1$, **real balances** evolve according to\n",
+ "\n",
+ "\n",
+ "$$\n",
+ "\\frac{m_{t+1}}{p_t} - \\frac{m_{t}}{p_{t-1}} \\frac{p_{t-1}}{p_t} = g\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "b_t - b_{t-1} R_{t-1} = g\n",
+ "$$ (eq:up_bmotion)\n",
+ "\n",
+ "where\n",
+ "\n",
+ "* $b_t = \\frac{m_{t+1}}{p_t}$ is real balances at the end of period $t$\n",
+ "* $R_{t-1} = \\frac{p_{t-1}}{p_t}$ is the gross rate of return on real balances held from $t-1$ to $t$\n",
+ "\n",
+ "The demand for real balances is \n",
+ "\n",
+ "$$\n",
+ "b_t = \\gamma_1 - \\gamma_2 R_t^{-1} . \n",
+ "$$ (eq:up_bdemand)\n",
+ "\n",
+ "where $\\gamma_1 > \\gamma_2 > 0$.\n",
+ "\n",
+ "## Monetary-Fiscal Policy\n",
+ "\n",
+ "To the basic model of {doc}`money_inflation`, we add inflation-indexed one-period government bonds as an additional way for the government to finance government expenditures. \n",
+ "\n",
+ "Let $\\widetilde R > 1$ be a time-invariant gross real rate of return on government one-period inflation-indexed bonds.\n",
+ "\n",
+ "With this additional source of funds, the government's budget constraint at time $t \\geq 0$ is now\n",
+ "\n",
+ "$$\n",
+ "B_t + \\frac{m_{t+1}}{p_t} = \\widetilde R B_{t-1} + \\frac{m_t}{p_t} + g\n",
+ "$$ \n",
+ "\n",
+ "\n",
+ "Just before the beginning of time $0$, the public owns $\\check m_0$ units of currency (measured in dollars)\n",
+ "and $\\widetilde R \\check B_{-1}$ units of one-period indexed bonds (measured in time $0$ goods); these two quantities are initial conditions set outside the model.\n",
+ "\n",
+ "Notice that $\\check m_0$ is a *nominal* quantity, being measured in dollars, while\n",
+ "$\\widetilde R \\check B_{-1}$ is a *real* quantity, being measured in time $0$ goods.\n",
+ "\n",
+ "\n",
+ "### Open market operations\n",
+ "\n",
+ "At time $0$, government can rearrange its portfolio of debts subject to the following constraint (on open-market operations):\n",
+ "\n",
+ "$$\n",
+ "\\widetilde R B_{-1} + \\frac{m_0}{p_0} = \\widetilde R \\check B_{-1} + \\frac{\\check m_0}{p_0}\n",
+ "$$\n",
+ "\n",
+ "or\n",
+ "\n",
+ "$$\n",
+ "B_{-1} - \\check B_{-1} = \\frac{1}{p_0 \\widetilde R} \\left( \\check m_0 - m_0 \\right) \n",
+ "$$ (eq:openmarketconstraint)\n",
+ "\n",
+ "This equation says that the government (e.g., the central bank) can *decrease* $m_0$ relative to \n",
+ "$\\check m_0$ by *increasing* $B_{-1}$ relative to $\\check B_{-1}$. \n",
+ "\n",
+ "This is a version of a standard constraint on a central bank's [**open market operations**](https://www.federalreserve.gov/monetarypolicy/openmarket.htm) in which it expands the stock of money by buying government bonds from the public. \n",
+ "\n",
+ "## An open market operation at $t=0$\n",
+ "\n",
+ "Following Sargent and Wallace {cite}`sargent1981`, we analyze consequences of a central bank policy that \n",
+ "uses an open market operation to lower the price level in the face of a persistent fiscal\n",
+ "deficit that takes the form of a positive $g$.\n",
+ "\n",
+ "Just before time $0$, the government chooses $(m_0, B_{-1})$ subject to constraint\n",
+ "{eq}`eq:openmarketconstraint`.\n",
+ "\n",
+ "For $t =0, 1, \\ldots, T-1$,\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "B_t & = \\widetilde R B_{t-1} + g \\cr\n",
+ "m_{t+1} & = m_0 \n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "while for $t \\geq T$,\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "B_t & = B_{T-1} \\cr\n",
+ "m_{t+1} & = m_t + p_t \\overline g\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "where \n",
+ "\n",
+ "$$\n",
+ "\\overline g = \\left[(\\tilde R -1) B_{T-1} + g \\right]\n",
+ "$$ (eq:overlineg)\n",
+ "\n",
+ "We want to compute an equilibrium $\\{p_t,m_t,b_t, R_t\\}_{t=0}$ sequence under this scheme for\n",
+ "running monetary and fiscal policies.\n",
+ "\n",
+ "Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $\\{g_t\\}_{t=0}^\\infty$ that must be financed by issuing to the public either money or interest bearing bonds.\n",
+ "\n",
+ "By **monetary policy** or **debt-management policy**, we mean the collection of actions that determine how the government divides its portfolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money).\n",
+ "\n",
+ "By an **open market operation**, we mean a government monetary policy action in which the government\n",
+ "(or its delegate, say, a central bank) either buys government bonds from the public for newly issued money, or sells bonds to the public and withdraws the money it receives from public circulation. \n",
+ "\n",
+ "## Algorithm (basic idea)\n",
+ "\n",
+ "\n",
+ "We work backwards from $t=T$ and first compute $p_T, R_u$ associated with the low-inflation, low-inflation-tax-rate stationary equilibrium in {doc}`money_inflation_nonlinear`.\n",
+ "\n",
+ "To start our description of our algorithm, it is useful to recall that a stationary rate of return\n",
+ "on currency $\\bar R$ solves the quadratic equation\n",
+ "\n",
+ "$$\n",
+ "-\\gamma_2 + (\\gamma_1 + \\gamma_2 - \\overline g) \\bar R - \\gamma_1 \\bar R^2 = 0\n",
+ "$$ (eq:up_steadyquadratic)\n",
+ "\n",
+ "Quadratic equation {eq}`eq:up_steadyquadratic` has two roots, $R_l < R_u < 1$.\n",
+ "\n",
+ "For reasons described at the end of {doc}`money_inflation`, we select the larger root $R_u$. \n",
+ "\n",
+ "\n",
+ "Next, we compute\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "R_T & = R_u \\cr\n",
+ "b_T & = \\gamma_1 - \\gamma_2 R_u^{-1} \\cr\n",
+ "p_T & = \\frac{m_0}{\\gamma_1 - \\overline g - \\gamma_2 R_u^{-1}}\n",
+ "\\end{aligned}\n",
+ "$$ (eq:LafferTstationary)\n",
+ "\n",
+ "\n",
+ "We can compute continuation sequences $\\{R_t, b_t\\}_{t=T+1}^\\infty$ of rates of return and real balances that are associated with an equilibrium by solving equation {eq}`eq:up_bmotion` and {eq}`eq:up_bdemand` sequentially for $t \\geq 1$:\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "b_t & = b_{t-1} R_{t-1} + \\overline g \\cr\n",
+ "R_t^{-1} & = \\frac{\\gamma_1}{\\gamma_2} - \\gamma_2^{-1} b_t \\cr\n",
+ "p_t & = R_t p_{t-1} \\cr\n",
+ " m_t & = b_{t-1} p_t \n",
+ "\\end{aligned}\n",
+ "$$\n",
+ " \n",
+ "\n",
+ "## Before time $T$ \n",
+ "\n",
+ "Define \n",
+ "\n",
+ "$$\n",
+ "\\lambda \\equiv \\frac{\\gamma_2}{\\gamma_1}.\n",
+ "$$\n",
+ "\n",
+ "Our restrictions that $\\gamma_1 > \\gamma_2 > 0$ imply that $\\lambda \\in [0,1)$.\n",
+ "\n",
+ "We want to compute\n",
+ "\n",
+ "$$ \n",
+ "\\begin{aligned}\n",
+ "p_0 & = \\gamma_1^{-1} \\left[ \\sum_{j=0}^\\infty \\lambda^j m_{j} \\right] \\cr\n",
+ "& = \\gamma_1^{-1} \\left[ \\sum_{j=0}^{T-1} \\lambda^j m_{0} + \\sum_{j=T}^\\infty \\lambda^j m_{1+j} \\right]\n",
+ "\\end{aligned}\n",
+ "$$\n",
+ "\n",
+ "Thus,\n",
+ "\n",
+ "$$\n",
+ "\\begin{aligned}\n",
+ "p_0 & = \\gamma_1^{-1} m_0 \\left\\{ \\frac{1 - \\lambda^T}{1-\\lambda} + \\frac{\\lambda^T}{R_u-\\lambda} \\right\\} \\cr\n",
+ "p_1 & = \\gamma_1^{-1} m_0 \\left\\{ \\frac{1 - \\lambda^{T-1}}{1-\\lambda} + \\frac{\\lambda^{T-1}}{R_u-\\lambda} \\right\\} \\cr\n",
+ "\\quad \\vdots & \\quad \\quad \\vdots \\cr\n",
+ "p_{T-1} & = \\gamma_1^{-1} m_0 \\left\\{ \\frac{1 - \\lambda}{1-\\lambda} + \\frac{\\lambda}{R_u-\\lambda} \\right\\} \\cr\n",
+ "p_T & = \\gamma_1^{-1} m_0 \\left\\{\\frac{1}{R_u-\\lambda} \\right\\}\n",
+ "\\end{aligned}\n",
+ "$$ (eq:allts)\n",
+ "\n",
+ "We can implement the preceding formulas by iterating on\n",
+ "\n",
+ "$$\n",
+ "p_t = \\gamma_1^{-1} m_0 + \\lambda p_{t+1}, \\quad t = T-1, T-2, \\ldots, 0\n",
+ "$$\n",
+ "\n",
+ "starting from \n",
+ "\n",
+ "$$\n",
+ "p_T = \\frac{m_0}{\\gamma_1 - \\overline g - \\gamma_2 R_u^{-1}} = \\gamma_1^{-1} m_0 \\left\\{\\frac{1}{R_u-\\lambda} \\right\\}\n",
+ "$$ (eq:pTformula)\n",
+ "\n",
+ "```{prf:remark}\n",
+ ":label: equivalence\n",
+ "We can verify the equivalence of the two formulas on the right sides of {eq}`eq:pTformula` by recalling that \n",
+ "$R_u$ is a root of the quadratic equation {eq}`eq:up_steadyquadratic` that determines steady state rates of return on currency.\n",
+ "```\n",
+ " \n",
+ "## Algorithm (pseudo code)\n",
+ "\n",
+ "Now let's describe a computational algorithm in more detail in the form of a description\n",
+ "that constitutes pseudo code because it approaches a set of instructions we could provide to a \n",
+ "Python coder.\n",
+ "\n",
+ "To compute an equilibrium, we deploy the following algorithm.\n",
+ "\n",
+ "```{prf:algorithm}\n",
+ "Given *parameters* include $g, \\check m_0, \\check B_{-1}, \\widetilde R >1, T $.\n",
+ "\n",
+ "We define a mapping from $p_0$ to $\\widehat p_0$ as follows.\n",
+ "\n",
+ "* Set $m_0$ and then compute $B_{-1}$ to satisfy the constraint on time $0$ **open market operations**\n",
+ "\n",
+ "$$\n",
+ "B_{-1}- \\check B_{-1} = \\frac{\\widetilde R}{p_0} \\left( \\check m_0 - m_0 \\right)\n",
+ "$$\n",
+ "\n",
+ "* Compute $B_{T-1}$ from\n",
+ "\n",
+ "$$\n",
+ "B_{T-1} = \\widetilde R^T B_{-1} + \\left( \\frac{1 - \\widetilde R^T}{1-\\widetilde R} \\right) g\n",
+ "$$\n",
+ "\n",
+ "* Compute \n",
+ "\n",
+ "$$\n",
+ "\\overline g = g + \\left[ \\tilde R - 1\\right] B_{T-1}\n",
+ "$$\n",
+ "\n",
+ "\n",
+ "\n",
+ "* Compute $R_u, p_T$ from formulas {eq}`eq:up_steadyquadratic` and {eq}`eq:LafferTstationary` above\n",
+ "\n",
+ "* Compute a new estimate of $p_0$, call it $\\widehat p_0$, from equation {eq}`eq:allts` above\n",
+ "\n",
+ "* Note that the preceding steps define a mapping\n",
+ "\n",
+ "$$\n",
+ "\\widehat p_0 = {\\mathcal S}(p_0)\n",
+ "$$\n",
+ "\n",
+ "* We seek a fixed point of ${\\mathcal S}$, i.e., a solution of $p_0 = {\\mathcal S}(p_0)$.\n",
+ "\n",
+ "* Compute a fixed point by iterating to convergence on the relaxation algorithm\n",
+ "\n",
+ "$$\n",
+ "p_{0,j+1} = (1-\\theta) {\\mathcal S}(p_{0,j}) + \\theta p_{0,j}, \n",
+ "$$\n",
+ "\n",
+ "where $\\theta \\in [0,1)$ is a relaxation parameter.\n",
+ "```\n",
+ "\n",
+ "## Example Calculations\n",
+ "\n",
+ "We'll set parameters of the model so that the steady state after time $T$ is initially the same\n",
+ "as in {doc}`money_inflation_nonlinear`\n",
+ "\n",
+ "In particular, we set $\\gamma_1=100, \\gamma_2 =50, g=3.0$. We set $m_0 = 100$ in that lecture,\n",
+ "but now the counterpart will be $M_T$, which is endogenous. \n",
+ "\n",
+ "As for new parameters, we'll set $\\tilde R = 1.01, \\check B_{-1} = 0, \\check m_0 = 105, T = 5$.\n",
+ "\n",
+ "We'll study a \"small\" open market operation by setting $m_0 = 100$.\n",
+ "\n",
+ "These parameter settings mean that just before time $0$, the \"central bank\" sells the public bonds in exchange for $\\check m_0 - m_0 = 5$ units of currency. \n",
+ "\n",
+ "That leaves the public with less currency but more government interest-bearing bonds.\n",
+ "\n",
+ "Since the public has less currency (its supply has diminished) it is plausible to anticipate that the price level at time $0$ will be driven downward.\n",
+ "\n",
+ "But that is not the end of the story, because this **open market operation** at time $0$ has consequences for future settings of $m_{t+1}$ and the gross-of-interest government deficit $\\bar g_t$. \n",
+ "\n",
+ "\n",
+ "Let's start with some imports:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "146031a5",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "from collections import namedtuple"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a0b7462b",
+ "metadata": {},
+ "source": [
+ "Now let's dive in and implement our pseudo code in Python."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8803613b",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Create a namedtuple that contains parameters\n",
+ "MoneySupplyModel = namedtuple(\"MoneySupplyModel\", \n",
+ " [\"γ1\", \"γ2\", \"g\",\n",
+ " \"R_tilde\", \"m0_check\", \"Bm1_check\",\n",
+ " \"T\"])\n",
+ "\n",
+ "def create_model(γ1=100, γ2=50, g=3.0,\n",
+ " R_tilde=1.01,\n",
+ " Bm1_check=0, m0_check=105,\n",
+ " T=5):\n",
+ " \n",
+ " return MoneySupplyModel(γ1=γ1, γ2=γ2, g=g,\n",
+ " R_tilde=R_tilde,\n",
+ " m0_check=m0_check, Bm1_check=Bm1_check,\n",
+ " T=T)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ad7fa03f",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "msm = create_model()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b20cbeb9",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def S(p0, m0, model):\n",
+ "\n",
+ " # unpack parameters\n",
+ " γ1, γ2, g = model.γ1, model.γ2, model.g\n",
+ " R_tilde = model.R_tilde\n",
+ " m0_check, Bm1_check = model.m0_check, model.Bm1_check\n",
+ " T = model.T\n",
+ "\n",
+ " # open market operation\n",
+ " Bm1 = 1 / (p0 * R_tilde) * (m0_check - m0) + Bm1_check\n",
+ "\n",
+ " # compute B_{T-1}\n",
+ " BTm1 = R_tilde ** T * Bm1 + ((1 - R_tilde ** T) / (1 - R_tilde)) * g\n",
+ "\n",
+ " # compute g bar\n",
+ " g_bar = g + (R_tilde - 1) * BTm1\n",
+ "\n",
+ " # solve the quadratic equation\n",
+ " Ru = np.roots((-γ1, γ1 + γ2 - g_bar, -γ2)).max()\n",
+ "\n",
+ " # compute p0\n",
+ " λ = γ2 / γ1\n",
+ " p0_new = (1 / γ1) * m0 * ((1 - λ ** T) / (1 - λ) + λ ** T / (Ru - λ))\n",
+ "\n",
+ " return p0_new"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "aea80c41",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def compute_fixed_point(m0, p0_guess, model, θ=0.5, tol=1e-6):\n",
+ "\n",
+ " p0 = p0_guess\n",
+ " error = tol + 1\n",
+ "\n",
+ " while error > tol:\n",
+ " p0_next = (1 - θ) * S(p0, m0, model) + θ * p0\n",
+ "\n",
+ " error = np.abs(p0_next - p0)\n",
+ " p0 = p0_next\n",
+ "\n",
+ " return p0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "d7023096",
+ "metadata": {},
+ "source": [
+ "Let's look at how price level $p_0$ in the stationary $R_u$ equilibrium depends on the initial\n",
+ "money supply $m_0$. \n",
+ "\n",
+ "Notice that the slope of $p_0$ as a function of $m_0$ is constant.\n",
+ "\n",
+ "This outcome indicates that our model verifies a quantity theory of money outcome,\n",
+ "something that Sargent and Wallace {cite}`sargent1981` purposefully built into their model to justify\n",
+ "the adjective *monetarist* in their title."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d024947f",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "m0_arr = np.arange(10, 110, 10)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "f7f4b0f0",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "plt.plot(m0_arr, [compute_fixed_point(m0, 1, msm) for m0 in m0_arr])\n",
+ "\n",
+ "plt.ylabel('$p_0$')\n",
+ "plt.xlabel('$m_0$')\n",
+ "\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "82be8e83",
+ "metadata": {},
+ "source": [
+ "Now let's write and implement code that lets us experiment with the time $0$ open market operation described earlier."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "2796e9f2",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def simulate(m0, model, length=15, p0_guess=1):\n",
+ "\n",
+ " # unpack parameters\n",
+ " γ1, γ2, g = model.γ1, model.γ2, model.g\n",
+ " R_tilde = model.R_tilde\n",
+ " m0_check, Bm1_check = model.m0_check, model.Bm1_check\n",
+ " T = model.T\n",
+ "\n",
+ " # (pt, mt, bt, Rt)\n",
+ " paths = np.empty((4, length))\n",
+ "\n",
+ " # open market operation\n",
+ " p0 = compute_fixed_point(m0, 1, model)\n",
+ " Bm1 = 1 / (p0 * R_tilde) * (m0_check - m0) + Bm1_check\n",
+ " BTm1 = R_tilde ** T * Bm1 + ((1 - R_tilde ** T) / (1 - R_tilde)) * g\n",
+ " g_bar = g + (R_tilde - 1) * BTm1\n",
+ " Ru = np.roots((-γ1, γ1 + γ2 - g_bar, -γ2)).max()\n",
+ "\n",
+ " λ = γ2 / γ1\n",
+ "\n",
+ " # t = 0\n",
+ " paths[0, 0] = p0\n",
+ " paths[1, 0] = m0\n",
+ "\n",
+ " # 1 <= t <= T\n",
+ " for t in range(1, T+1, 1):\n",
+ " paths[0, t] = (1 / γ1) * m0 * \\\n",
+ " ((1 - λ ** (T - t)) / (1 - λ)\n",
+ " + (λ ** (T - t) / (Ru - λ)))\n",
+ " paths[1, t] = m0\n",
+ "\n",
+ " # t > T\n",
+ " for t in range(T+1, length):\n",
+ " paths[0, t] = paths[0, t-1] / Ru\n",
+ " paths[1, t] = paths[1, t-1] + paths[0, t] * g_bar\n",
+ "\n",
+ " # Rt = pt / pt+1\n",
+ " paths[3, :T] = paths[0, :T] / paths[0, 1:T+1]\n",
+ " paths[3, T:] = Ru\n",
+ "\n",
+ " # bt = γ1 - γ2 / Rt\n",
+ " paths[2, :] = γ1 - γ2 / paths[3, :]\n",
+ "\n",
+ " return paths"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "1d2acafb",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def plot_path(m0_arr, model, length=15):\n",
+ "\n",
+ " fig, axs = plt.subplots(2, 2, figsize=(8, 5))\n",
+ " titles = ['$p_t$', '$m_t$', '$b_t$', '$R_t$']\n",
+ " \n",
+ " for m0 in m0_arr:\n",
+ " paths = simulate(m0, model, length=length)\n",
+ " for i, ax in enumerate(axs.flat):\n",
+ " ax.plot(paths[i])\n",
+ " ax.set_title(titles[i])\n",
+ " \n",
+ " axs[0, 1].hlines(model.m0_check, 0, length, color='r', linestyle='--')\n",
+ " axs[0, 1].text(length * 0.8, model.m0_check * 0.9, r'$\\check{m}_0$')\n",
+ " plt.show()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "b4ddb909",
+ "metadata": {
+ "mystnb": {
+ "figure": {
+ "caption": "Unpleasant Arithmetic",
+ "name": "fig:unpl1"
+ }
+ }
+ },
+ "outputs": [],
+ "source": [
+ "plot_path([80, 100], msm)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2d04abbb",
+ "metadata": {},
+ "source": [
+ "{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of Sargent and Wallace {cite}`sargent1981`.\n",
+ "\n",
+ "* An open market operation that reduces the supply of money at time $t=0$ reduces the price level at time $t=0$\n",
+ "\n",
+ "* The lower is the post-open-market-operation money supply at time $0$, lower is the price level at time $0$.\n",
+ "\n",
+ "* An open market operation that reduces the post open market operation money supply at time $0$ also *lowers* the rate of return on money $R_u$ at times $t \\geq T$ because it brings a higher gross of interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \\geq T$.\n",
+ "\n",
+ "* $R$ is important in the context of maintaining monetary stability and addressing the consequences of increased inflation due to government deficits. Thus, a larger $R$ might be chosen to mitigate the negative impacts on the real rate of return caused by inflation."
+ ]
+ }
+ ],
+ "metadata": {
+ "jupytext": {
+ "text_representation": {
+ "extension": ".md",
+ "format_name": "myst",
+ "format_version": 0.13,
+ "jupytext_version": "1.14.1"
+ }
+ },
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "source_map": [
+ 12,
+ 326,
+ 330,
+ 334,
+ 352,
+ 356,
+ 384,
+ 397,
+ 409,
+ 413,
+ 420,
+ 424,
+ 471,
+ 488,
+ 496
+ ]
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/lectures/unpleasant.md b/_sources/unpleasant.md
similarity index 100%
rename from lectures/unpleasant.md
rename to _sources/unpleasant.md
diff --git a/_sources/zreferences.ipynb b/_sources/zreferences.ipynb
new file mode 100644
index 000000000..9bbaeab0c
--- /dev/null
+++ b/_sources/zreferences.ipynb
@@ -0,0 +1,34 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "ba50e9d1",
+ "metadata": {},
+ "source": [
+ "(references)=\n",
+ "# References\n",
+ "\n",
+ "```{bibliography} _static/quant-econ.bib\n",
+ "```"
+ ]
+ }
+ ],
+ "metadata": {
+ "jupytext": {
+ "text_representation": {
+ "extension": ".md",
+ "format_name": "myst"
+ }
+ },
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "source_map": [
+ 10
+ ]
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
\ No newline at end of file
diff --git a/lectures/zreferences.md b/_sources/zreferences.md
similarity index 100%
rename from lectures/zreferences.md
rename to _sources/zreferences.md
diff --git a/_sphinx_design_static/design-tabs.js b/_sphinx_design_static/design-tabs.js
new file mode 100644
index 000000000..b25bd6a4f
--- /dev/null
+++ b/_sphinx_design_static/design-tabs.js
@@ -0,0 +1,101 @@
+// @ts-check
+
+// Extra JS capability for selected tabs to be synced
+// The selection is stored in local storage so that it persists across page loads.
+
+/**
+ * @type {Record\n",
+ " \n",
+ "Short
+ */ + .o-tooltip--left { + position: relative; + } + + .o-tooltip--left:after { + opacity: 0; + visibility: hidden; + position: absolute; + content: attr(data-tooltip); + padding: .2em; + font-size: .8em; + left: -.2em; + background: grey; + color: white; + white-space: nowrap; + z-index: 2; + border-radius: 2px; + transform: translateX(-102%) translateY(0); + transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); +} + +.o-tooltip--left:hover:after { + display: block; + opacity: 1; + visibility: visible; + transform: translateX(-100%) translateY(0); + transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1); + transition-delay: .5s; +} + +/* By default the copy button shouldn't show up when printing a page */ +@media print { + button.copybtn { + display: none; + } +} diff --git a/_static/copybutton.js b/_static/copybutton.js new file mode 100644 index 000000000..2ea7ff3e2 --- /dev/null +++ b/_static/copybutton.js @@ -0,0 +1,248 @@ +// Localization support +const messages = { + 'en': { + 'copy': 'Copy', + 'copy_to_clipboard': 'Copy to clipboard', + 'copy_success': 'Copied!', + 'copy_failure': 'Failed to copy', + }, + 'es' : { + 'copy': 'Copiar', + 'copy_to_clipboard': 'Copiar al portapapeles', + 'copy_success': '¡Copiado!', + 'copy_failure': 'Error al copiar', + }, + 'de' : { + 'copy': 'Kopieren', + 'copy_to_clipboard': 'In die Zwischenablage kopieren', + 'copy_success': 'Kopiert!', + 'copy_failure': 'Fehler beim Kopieren', + }, + 'fr' : { + 'copy': 'Copier', + 'copy_to_clipboard': 'Copier dans le presse-papier', + 'copy_success': 'Copié !', + 'copy_failure': 'Échec de la copie', + }, + 'ru': { + 'copy': 'Скопировать', + 'copy_to_clipboard': 'Скопировать в буфер', + 'copy_success': 'Скопировано!', + 'copy_failure': 'Не удалось скопировать', + }, + 'zh-CN': { + 'copy': '复制', + 'copy_to_clipboard': '复制到剪贴板', + 'copy_success': '复制成功!', + 'copy_failure': '复制失败', + }, + 'it' : { + 'copy': 'Copiare', + 'copy_to_clipboard': 'Copiato negli appunti', + 'copy_success': 'Copiato!', + 'copy_failure': 'Errore durante la copia', + } +} + +let locale = 'en' +if( document.documentElement.lang !== undefined + && messages[document.documentElement.lang] !== undefined ) { + locale = document.documentElement.lang +} + +let doc_url_root = DOCUMENTATION_OPTIONS.URL_ROOT; +if (doc_url_root == '#') { + doc_url_root = ''; +} + +/** + * SVG files for our copy buttons + */ +let iconCheck = `` + +// If the user specified their own SVG use that, otherwise use the default +let iconCopy = ``; +if (!iconCopy) { + iconCopy = `` +} + +/** + * Set up copy/paste for code blocks + */ + +const runWhenDOMLoaded = cb => { + if (document.readyState != 'loading') { + cb() + } else if (document.addEventListener) { + document.addEventListener('DOMContentLoaded', cb) + } else { + document.attachEvent('onreadystatechange', function() { + if (document.readyState == 'complete') cb() + }) + } +} + +const codeCellId = index => `codecell${index}` + +// Clears selected text since ClipboardJS will select the text when copying +const clearSelection = () => { + if (window.getSelection) { + window.getSelection().removeAllRanges() + } else if (document.selection) { + document.selection.empty() + } +} + +// Changes tooltip text for a moment, then changes it back +// We want the timeout of our `success` class to be a bit shorter than the +// tooltip and icon change, so that we can hide the icon before changing back. +var timeoutIcon = 2000; +var timeoutSuccessClass = 1500; + +const temporarilyChangeTooltip = (el, oldText, newText) => { + el.setAttribute('data-tooltip', newText) + el.classList.add('success') + // Remove success a little bit sooner than we change the tooltip + // So that we can use CSS to hide the copybutton first + setTimeout(() => el.classList.remove('success'), timeoutSuccessClass) + setTimeout(() => el.setAttribute('data-tooltip', oldText), timeoutIcon) +} + +// Changes the copy button icon for two seconds, then changes it back +const temporarilyChangeIcon = (el) => { + el.innerHTML = iconCheck; + setTimeout(() => {el.innerHTML = iconCopy}, timeoutIcon) +} + +const addCopyButtonToCodeCells = () => { + // If ClipboardJS hasn't loaded, wait a bit and try again. This + // happens because we load ClipboardJS asynchronously. + if (window.ClipboardJS === undefined) { + setTimeout(addCopyButtonToCodeCells, 250) + return + } + + // Add copybuttons to all of our code cells + const COPYBUTTON_SELECTOR = 'div.highlight pre'; + const codeCells = document.querySelectorAll(COPYBUTTON_SELECTOR) + codeCells.forEach((codeCell, index) => { + const id = codeCellId(index) + codeCell.setAttribute('id', id) + + const clipboardButton = id => + `` + codeCell.insertAdjacentHTML('afterend', clipboardButton(id)) + }) + +function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string +} + +/** + * Removes excluded text from a Node. + * + * @param {Node} target Node to filter. + * @param {string} exclude CSS selector of nodes to exclude. + * @returns {DOMString} Text from `target` with text removed. + */ +function filterText(target, exclude) { + const clone = target.cloneNode(true); // clone as to not modify the live DOM + if (exclude) { + // remove excluded nodes + clone.querySelectorAll(exclude).forEach(node => node.remove()); + } + return clone.innerText; +} + +// Callback when a copy button is clicked. Will be passed the node that was clicked +// should then grab the text and replace pieces of text that shouldn't be used in output +function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") { + var regexp; + var match; + + // Do we check for line continuation characters and "HERE-documents"? + var useLineCont = !!lineContinuationChar + var useHereDoc = !!hereDocDelim + + // create regexp to capture prompt and remaining line + if (isRegexp) { + regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') + } else { + regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') + } + + const outputLines = []; + var promptFound = false; + var gotLineCont = false; + var gotHereDoc = false; + const lineGotPrompt = []; + for (const line of textContent.split('\n')) { + match = line.match(regexp) + if (match || gotLineCont || gotHereDoc) { + promptFound = regexp.test(line) + lineGotPrompt.push(promptFound) + if (removePrompts && promptFound) { + outputLines.push(match[2]) + } else { + outputLines.push(line) + } + gotLineCont = line.endsWith(lineContinuationChar) & useLineCont + if (line.includes(hereDocDelim) & useHereDoc) + gotHereDoc = !gotHereDoc + } else if (!onlyCopyPromptLines) { + outputLines.push(line) + } else if (copyEmptyLines && line.trim() === '') { + outputLines.push(line) + } + } + + // If no lines with the prompt were found then just use original lines + if (lineGotPrompt.some(v => v === true)) { + textContent = outputLines.join('\n'); + } + + // Remove a trailing newline to avoid auto-running when pasting + if (textContent.endsWith("\n")) { + textContent = textContent.slice(0, -1) + } + return textContent +} + + +var copyTargetText = (trigger) => { + var target = document.querySelector(trigger.attributes['data-clipboard-target'].value); + + // get filtered text + let exclude = '.linenos'; + + let text = filterText(target, exclude); + return formatCopyText(text, '', false, true, true, true, '', '') +} + + // Initialize with a callback so we can modify the text before copy + const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText}) + + // Update UI with error/success messages + clipboard.on('success', event => { + clearSelection() + temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_success']) + temporarilyChangeIcon(event.trigger) + }) + + clipboard.on('error', event => { + temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_failure']) + }) +} + +runWhenDOMLoaded(addCopyButtonToCodeCells) \ No newline at end of file diff --git a/_static/copybutton_funcs.js b/_static/copybutton_funcs.js new file mode 100644 index 000000000..dbe1aaad7 --- /dev/null +++ b/_static/copybutton_funcs.js @@ -0,0 +1,73 @@ +function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string +} + +/** + * Removes excluded text from a Node. + * + * @param {Node} target Node to filter. + * @param {string} exclude CSS selector of nodes to exclude. + * @returns {DOMString} Text from `target` with text removed. + */ +export function filterText(target, exclude) { + const clone = target.cloneNode(true); // clone as to not modify the live DOM + if (exclude) { + // remove excluded nodes + clone.querySelectorAll(exclude).forEach(node => node.remove()); + } + return clone.innerText; +} + +// Callback when a copy button is clicked. Will be passed the node that was clicked +// should then grab the text and replace pieces of text that shouldn't be used in output +export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") { + var regexp; + var match; + + // Do we check for line continuation characters and "HERE-documents"? + var useLineCont = !!lineContinuationChar + var useHereDoc = !!hereDocDelim + + // create regexp to capture prompt and remaining line + if (isRegexp) { + regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)') + } else { + regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)') + } + + const outputLines = []; + var promptFound = false; + var gotLineCont = false; + var gotHereDoc = false; + const lineGotPrompt = []; + for (const line of textContent.split('\n')) { + match = line.match(regexp) + if (match || gotLineCont || gotHereDoc) { + promptFound = regexp.test(line) + lineGotPrompt.push(promptFound) + if (removePrompts && promptFound) { + outputLines.push(match[2]) + } else { + outputLines.push(line) + } + gotLineCont = line.endsWith(lineContinuationChar) & useLineCont + if (line.includes(hereDocDelim) & useHereDoc) + gotHereDoc = !gotHereDoc + } else if (!onlyCopyPromptLines) { + outputLines.push(line) + } else if (copyEmptyLines && line.trim() === '') { + outputLines.push(line) + } + } + + // If no lines with the prompt were found then just use original lines + if (lineGotPrompt.some(v => v === true)) { + textContent = outputLines.join('\n'); + } + + // Remove a trailing newline to avoid auto-running when pasting + if (textContent.endsWith("\n")) { + textContent = textContent.slice(0, -1) + } + return textContent +} diff --git a/_static/design-tabs.js b/_static/design-tabs.js new file mode 100644 index 000000000..b25bd6a4f --- /dev/null +++ b/_static/design-tabs.js @@ -0,0 +1,101 @@ +// @ts-check + +// Extra JS capability for selected tabs to be synced +// The selection is stored in local storage so that it persists across page loads. + +/** + * @type {Record